This is an archive of email messages concerning the Geometric Langlands Seminar for 2012-13.
As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. First meetings: October 4 (Thursday) and October 8 (Monday). The seminar will begin with a talk by Beilinson followed by a series of talks by Nick Rozenblyum (NWU). The latter will be devoted to a new approach to the foundations of D-module theory developed by Gaitsgory and Rozenblyum. Beilinson's talk is intended to be a kind of introduction to those by Rozenblyum.
Thursday (October 4), 4:30 p.m, room E 206.
Alexander Beilinson. Crystals, D-modules, and derived algebraic geometry.
Abstract
This is an introduction to a series of talks of Nick Rosenblum on his
foundational work with Dennis Gaitsgory that establishes the basic
D-module
functoriality in the context of derived algebraic geometry (hence for
arbitrary singular algebraic varieties) over a field of characteristic 0.
I will discuss the notion of crystals and de Rham coefficients that goes
back to Grothendieck, the derived D-module functoriality for smooth
varieties (due to Bernstein and Kashiwara), and some basic ideas of the
Gaitsgory-Rosenblum theory. No previous knowledge of the above subjects is
needed.
Monday (October 8), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU).
Duality and D-modules via derived algebraic geometry.
Abstract
I will describe joint work with D. Gaitsgory formulating the theory of
D-modules using derived algebraic geometry. I will begin with an overview
of Grothendieck-Serre duality in derived algebraic geometry via the
formalism of ind-coherent sheaves. The theory of D-modules will be built
as an extension of this theory.
A key player in the story is the deRham stack, introduced by Simpson in
the context of nonabelian Hodge theory. It is a convenient formulation of
Gorthendieck's theory of crystals in characteristic 0. I will explain its
construction and basic properties. The category of D-modules is defined
as sheaves in the deRham stack. This construction has a number of
benefits; for instance, Kashiwara's Lemma and h-descent are easy
consequences of the definition. I will also explain how this approach
compares to more familiar definitions.
Thursday (October 11), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU).
Duality and D-modules via derived algebraic geometry. II
Abstract
> I will describe joint work with D. Gaitsgory formulating the theory of
> D-modules using derived algebraic geometry. I will begin with an overview
> of Grothendieck-Serre duality in derived algebraic geometry via the
> formalism of ind-coherent sheaves. The theory of D-modules will be built
> as an extension of this theory.
>
> A key player in the story is the deRham stack, introduced by Simpson in
> the context of nonabelian Hodge theory. It is a convenient formulation of
> Gorthendieck's theory of crystals in characteristic 0. I will explain its
> construction and basic properties. The category of D-modules is defined
> as sheaves in the deRham stack. This construction has a number of
> benefits; for instance, Kashiwara's Lemma and h-descent are easy
> consequences of the definition. I will also explain how this approach
> compares to more familiar definitions.
>
>
>
>
>
>
>
>
No seminar on Monday. Nick will continue next Thursday: Thursday (October 18), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. III > Abstract > > I will describe joint work with D. Gaitsgory formulating the theory of > D-modules using derived algebraic geometry. I will begin with an > overview of Grothendieck-Serre duality in derived algebraic geometry via > the formalism of ind-coherent sheaves. The theory of D-modules will be > built as an extension of this theory. > > A key player in the story is the deRham stack, introduced by Simpson in > the context of nonabelian Hodge theory. It is a convenient formulation > of Grothendieck's theory of crystals in characteristic 0. I will explain > its construction and basic properties. The category of D-modules is > defined as sheaves in the deRham stack. This construction has a number of > benefits; for instance, Kashiwara's Lemma and h-descent are easy > consequences of the definition. I will also explain how this approach > compares to more familiar definitions.
Thursday (October 18), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. III > Abstract > > I will describe joint work with D. Gaitsgory formulating the theory of > D-modules using derived algebraic geometry. I will begin with an > overview of Grothendieck-Serre duality in derived algebraic geometry via > the formalism of ind-coherent sheaves. The theory of D-modules will be > built as an extension of this theory. > > A key player in the story is the deRham stack, introduced by Simpson in > the context of nonabelian Hodge theory. It is a convenient formulation > of Grothendieck's theory of crystals in characteristic 0. I will explain > its construction and basic properties. The category of D-modules is > defined as sheaves in the deRham stack. This construction has a number of > benefits; for instance, Kashiwara's Lemma and h-descent are easy > consequences of the definition. I will also explain how this approach > compares to more familiar definitions.
Monday (October 22), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. IV.
No seminar until Mitya Boyarchenko's talk on Nov 8. (So we have plenty of time to think about Nick's talks!) Please note Sarnak's Albert lectures on Oct 31 (Wednesday), Nov 1 (Thursday), Nov 2 (Friday), see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml
Peter Sarnak's Albert lectures have been moved to Nov 7-9, see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml As announced before, Mitya Boyarchenko will speak on Nov 8 (Thursday). On Nov 12 (Monday) Jonathan Barlev will begin his series of talks on the spaces of rational maps.
No seminar tomorrow (Monday).
The title&abstract of Mitya Boyarchenko's Thursday talks are below.
Please note Sarnak's Albert lectures on
"Randomness in Number Theory"
on Wednesday, Thursday, and Friday, see
http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml
*************
Thursday (Nov 8), 4:30 p.m, room E 206.
Mitya Boyarchenko (University of Michigan)
New geometric structures in the local Langlands program.
Abstract
The problem of explicitly constructing the local Langlands
correspondence for GL_n(K), where K is a p-adic field, contains as an
important special case the problem of constructing automorphic
induction (or "twisted parabolic induction") from certain
1-dimensional characters of L^* (where L is a given Galois extension of K
of degree n) to irreducible supercuspidal representations of
GL_n(K). Already in 1979 Lusztig proposed a very elegant, but still
conjectural, geometric construction of twisted parabolic induction for
unramified maximal tori in arbitrary reductive p-adic groups. An
analysis of Lusztig's construction and of the Lubin-Tate tower of K leads
to interesting new varieties that provide an analogue of
Deligne-Lusztig theory for certain families of unipotent groups over
finite fields. I will describe the known examples of this phenomenon and
their relationship to the local Langlands correspondence. All the
necessary background will be provided. Part of the talk will be based on
joint work with Jared Weinstein (Boston University).
Thursday (Nov 8), 4:30 p.m, room E 206.
Mitya Boyarchenko (University of Michigan)
New geometric structures in the local Langlands program.
(Sarnak's second Albert lecture is at 3 p.m., so you can easily attend
both lectures).
*************
Abstract
The problem of explicitly constructing the local Langlands
correspondence for GL_n(K), where K is a p-adic field, contains as an
important special case the problem of constructing automorphic
induction (or "twisted parabolic induction") from certain
1-dimensional characters of L^* (where L is a given Galois extension of K
of degree n) to irreducible supercuspidal representations of
GL_n(K). Already in 1979 Lusztig proposed a very elegant, but still
conjectural, geometric construction of twisted parabolic induction for
unramified maximal tori in arbitrary reductive p-adic groups. An
analysis of Lusztig's construction and of the Lubin-Tate tower of K leads
to interesting new varieties that provide an analogue of
Deligne-Lusztig theory for certain families of unipotent groups over
finite fields. I will describe the known examples of this phenomenon and
their relationship to the local Langlands correspondence. All the
necessary background will be provided. Part of the talk will be based on
joint work with Jared Weinstein (Boston University).
Monday (Nov 12), 4:30 p.m, room E 206.
Jonathan Barlev. Models for spaces of rational maps
Abstract
I will discuss the equivalence between three different models for spaces
of rational maps in algebraic geometry. In particular, I will explain the
relation between spaces of quasi-maps and the model for the space of
rational maps which Gaitsgory uses in his recent contractibility theorem.
Categories of D-modules on spaces of rational maps arise in the context of
the geometric Langlands program. However, as such spaces are not
representable by (ind-)schemes, the construction of such categories relies
on the general theory presented in Nick Rozenblyum's talks. I will explain
how each of the different models for these spaces exhibit different
properties of their categories of D-modules.
Thursday (Nov 15), 4:30 p.m, room E 206.
Jonathan Barlev. Models for spaces of rational maps. II.
Abstract
I will discuss the equivalence between three different models for spaces
of rational maps in algebraic geometry. In particular, I will explain the
relation between spaces of quasi-maps and the model for the space of
rational maps which Gaitsgory uses in his recent contractibility theorem.
Categories of D-modules on spaces of rational maps arise in the context of
the geometric Langlands program. However, as such spaces are not
representable by (ind-)schemes, the construction of such categories relies
on the general theory presented in Nick Rozenblyum's talks. I will explain
how each of the different models for these spaces exhibit different
properties of their categories of D-modules.
No seminar until Thanksgiving.
John Francis (NWU) will give his first talk after Thanksgiving
(probably on Thursday).
*******
Attached is a proof of the contractibility statement in the classical
topology (over the complex numbers). Please check.
I make there two additional assumptions, which are not really necessary:
(a) I assume that the target variety equals {affine space}-{hypersurface}.
This implies the statement in the more general setting considered at the
seminar (when the target variety is connected and locally isomorphic to an
affine space). One uses here the following fact: if a topological space is
covered by open sets so that all finite intersections of these subsets are
contractible then the whole space is contractible.
(b) I assume that K is the field of rational functions. This immediately
implies the statement for any finite extension of K. To see this, note
that
if K' is an extension of K of degree n then (K')^m =K^{mn}. The scientific
name for this is "Weil restriction of scalars".
Attachment:
Contractibility.pdf
Description: Adobe PDF document
No seminar on Monday (Nov 26).
Thursday (Nov 29), 4:30 p.m, room E 206.
John Francis (NWU). Factorization homology of topological manifolds.
Abstract
Factorization homology, a.k.a. topological chiral homology, of Lurie, is a
type of homology theory for n-manifolds whose system of coefficients is
given by an n-disk, or E_n-, algebra. It was formulated as a topological
analogue of the homology of the algebro-geometric factorization algebras
of Beilinson & Drinfeld, and it generalizes previous work in topology of
Salvatore and Segal. Factorization homology is characterized by a
generalization of the Eilenberg-Steenrod axioms. I'll use this to give a
short proof of nonabelian Poincare duality and then discuss other
calculations, including factorization homology with coefficients in
enveloping algebras of Lie algebras -- a topological analogue of Beilinson
& Drinfeld's description of chiral homology of chiral enveloping algebras.
Thursday (Nov 29), 4:30 p.m, room E 206.
John Francis (NWU). Factorization homology of topological manifolds.
Abstract
Factorization homology, a.k.a. topological chiral homology, of Lurie, is a
type of homology theory for n-manifolds whose system of coefficients is
given by an n-disk, or E_n-, algebra. It was formulated as a topological
analogue of the homology of the algebro-geometric factorization algebras
of Beilinson & Drinfeld, and it generalizes previous work in topology of
Salvatore and Segal. Factorization homology is characterized by a
generalization of the Eilenberg-Steenrod axioms. I'll use this to give a
short proof of nonabelian Poincare duality and then discuss other
calculations, including factorization homology with coefficients in
enveloping algebras of Lie algebras -- a topological analogue of Beilinson
& Drinfeld's description of chiral homology of chiral enveloping algebras.
Thursday (Dec 6), 4:30 p.m, room E 206.
John Francis (NWU). Factorization homology of topological manifolds.II.
Abstract
Factorization homology, a.k.a. topological chiral homology, of Lurie, is a
type of homology theory for n-manifolds whose system of coefficients is
given by an n-disk, or E_n-, algebra. It was formulated as a topological
analogue of the homology of the algebro-geometric factorization algebras
of Beilinson & Drinfeld, and it generalizes previous work in topology of
Salvatore and Segal. Factorization homology is characterized by a
generalization of the Eilenberg-Steenrod axioms. I'll use this to give a
short proof of nonabelian Poincare duality and then discuss other
calculations, including factorization homology with coefficients in
enveloping algebras of Lie algebras -- a topological analogue of Beilinson
& Drinfeld's description of chiral homology of chiral enveloping algebras.
Thursday (Dec 6), 4:30 p.m, room E 206.
John Francis (NWU). Factorization homology of topological manifolds.II.
Abstract
Factorization homology, a.k.a. topological chiral homology, of Lurie, is a
type of homology theory for n-manifolds whose system of coefficients is
given by an n-disk, or E_n-, algebra. It was formulated as a topological
analogue of the homology of the algebro-geometric factorization algebras
of Beilinson & Drinfeld, and it generalizes previous work in topology of
Salvatore and Segal. Factorization homology is characterized by a
generalization of the Eilenberg-Steenrod axioms. I'll use this to give a
short proof of nonabelian Poincare duality and then discuss other
calculations, including factorization homology with coefficients in
enveloping algebras of Lie algebras -- a topological analogue of Beilinson
& Drinfeld's description of chiral homology of chiral enveloping algebras.
No more meetings of the Geometric Langlands seminar this quarter.
The geometric Langlands seminar does not meet this week. Next Monday (January 14) Beilinson will give an introductory talk on topological cyclic homology, to be followed by T.Goodwillie's talk on the same subject on Thursday January 17. On Jan 21 and 24 Nick Rozenblyum will explain Kevin Costello's approach to the Witten genus. Next speakers: Bhargav Bhatt (Jan 28), Jared Weinstein: February 4,5,7.
Monday (January 14), 4:30 p.m, room E 206.
Alexander Beilinson. An introduction to Goodwillie's talk on topological
cyclic homology.
[Presumably, in his Thursday talk Goodwillie will explain several ways of
looking at topological cyclic homology.]
Abstract
My talk is intended to serve as an introduction to T.Goodwillie's talk on
Thursday January 17. No prior knowledge of the subject is assumed.
A recent article by Bloch, Esnault, and Kerz about p-adic deformations of
algebraic cycles uses topological cyclic homology (TCH) as a principal, if
hidden, tool. I will try to explain the main features of TCH theory and
discuss the relation of TCH to classical cyclic homology as motivated by
the Bloch-Esnault-Kerz work and explained to me by V.Angeltveit and
N.Rozenblum. No prior knowledge of the subject is assumed.
Below are:
(i) information on Goodwillie's Thursday talk;
(ii) a link to an article by Peter May.
*******
Thursday (January 17), 4:30 p.m, room E 206.
Thomas Goodwillie (Brown University). On topological cyclic homology.
Abstract
The cyclotomic trace is an important map from algebraic K-theory whose
target is a kind of topological cyclic homology. Rationally it can be
defined purely algebraically, but integrally its definition uses
equivariant stable homotopy theory. I will look at this topic from
several points of view. In particular it is interesting to look at the
cyclotomic trace in the case of Waldhausen K-theory, where it leads to
equivariant constructions on loops in a manifold.
******
Here is the link to Peter May's notes for a 1997 talk:
http://www.math.uchicago.edu/~may/TALKS/THHTC.pdf
The talk was before anyone was using orthogonal spectra
(although in fact Peter May first defined them in a 1980 paper).
Monday (January 21), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the
Witten genus.
Abstract
We will describe a formal version of nonabelian duality in derived
algebraic geometry, using the Beilinson-Drinfeld theory of chiral
algebras. This provides a local-to-global approach to the study of a
certain class of moduli spaces -- such as mapping spaces, the moduli space
of curves and the moduli space of principal G-bundles. In this context, we
will describe a quantization procedure and the associated theory of
Feynman integration. As an application, we obtain an algebro-geometric
version of Costello's construction of the Witten genus.
Thursday (January 24), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the
Witten genus. II.
Abstract
We will describe a formal version of nonabelian duality in derived
algebraic geometry, using the Beilinson-Drinfeld theory of chiral
algebras. This provides a local-to-global approach to the study of a
certain class of moduli spaces -- such as mapping spaces, the moduli space
of curves and the moduli space of principal G-bundles. In this context, we
will describe a quantization procedure and the associated theory of
Feynman integration. As an application, we obtain an algebro-geometric
version of Costello's construction of the Witten genus.
Monday (Jan 28), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the
Witten genus. III.
Abstract
We will describe a formal version of nonabelian duality in derived
algebraic geometry, using the Beilinson-Drinfeld theory of chiral
algebras. This provides a local-to-global approach to the study of a
certain class of moduli spaces -- such as mapping spaces, the moduli space
of curves and the moduli space of principal G-bundles. In this context, we
will describe a quantization procedure and the associated theory of
Feynman integration. As an application, we obtain an algebro-geometric
version of Costello's construction of the Witten genus.
I am resending this message, just in case.
*******
Monday (Jan 28), 4:30 p.m, room E 206.
Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the
Witten genus. III.
Abstract
We will describe a formal version of nonabelian duality in derived
algebraic geometry, using the Beilinson-Drinfeld theory of chiral
algebras. This provides a local-to-global approach to the study of a
certain class of moduli spaces -- such as mapping spaces, the moduli space
of curves and the moduli space of principal G-bundles. In this context, we
will describe a quantization procedure and the associated theory of
Feynman integration. As an application, we obtain an algebro-geometric
version of Costello's construction of the Witten genus.
No seminar on Thursday this week. ****** Next week Jared Weinstein (Boston University) will speak at the Langlands seminar on Monday and Thursday. He will also speak at the Number Theory seminar on Tuesday. To the best of my knowledge, his talks will be related to the following works: http://arxiv.org/abs/1207.6424 http://arxiv.org/abs/1211.6357 More details will be announced later.
Monday (Feb 4), 4:30 p.m, room E 206.
Jared Weinstein (Boston University). Moduli of formal groups with infinite
level structure. I.
Prof. Weinstein will also speak at the Langlands seminar on Thursday and
at the Number Theory seminar on Tuesday, see
http://www.math.uchicago.edu/~reduzzi/NTseminar/
Abstract
A formal group is a bi-variate formal power series which mimics the
behavior of an abelian group. More generally one can talk about formal
$O$-modules, where $O$ is any ring.
Suppose $K$ is a local nonarchimedean field with ring of integers $O$ and
residue field $k$. For each $n$, there is up to isomorphism a unique
formal $O$-module $H$ of height $n$ over the algebraic closure of $k$. In
1974, Drinfeld introduced an ascending family of regular local rings $A_m$
which parameterize deformations of $H$ with level $m$ structure. These
rings are implicated in the proof by Harris and Taylor of the local
Langlands correspondence for GL_n(K). In this talk, we will discuss the
ring $A$ obtained by completing the union of the $A_m$. It turns out that
this ring has a very explicit description -- despite not being noetherian,
it is somehow simpler than any of the finite level rings $A_m$. These
observations generalize to other deformation spaces of p-divisible groups
(joint work with Peter Scholze), and suggest the usefulness of working at
infinite level in the context of other arithmetic moduli problems.
Thursday (Feb 7), 4:30 p.m, room E 206.
Jared Weinstein (Boston University). Moduli of formal groups with infinite
level structure. II.
Abstract
A formal group is a bi-variate formal power series which mimics the
behavior of an abelian group. More generally one can talk about formal
$O$-modules, where $O$ is any ring.
Suppose $K$ is a local nonarchimedean field with ring of integers $O$ and
residue field $k$. For each $n$, there is up to isomorphism a unique
formal $O$-module $H$ of height $n$ over the algebraic closure of $k$. In
1974, Drinfeld introduced an ascending family of regular local rings $A_m$
which parameterize deformations of $H$ with level $m$ structure. These
rings are implicated in the proof by Harris and Taylor of the local
Langlands correspondence for GL_n(K). In this talk, we will discuss the
ring $A$ obtained by completing the union of the $A_m$. It turns out that
this ring has a very explicit description -- despite not being noetherian,
it is somehow simpler than any of the finite level rings $A_m$. These
observations generalize to other deformation spaces of p-divisible groups
(joint work with Peter Scholze), and suggest the usefulness of working at
infinite level in the context of other arithmetic moduli problems.
Monday (Feb 11), 4:30 p.m, room E 206.
David Kazhdan (Hebrew University).
Minimal representations of simply-laced reductive groups.
Abstract
For any local field F the Weil representation is a representation of
M(2n,f), the double cover of the group Sp(2n,F); this remarkable
representation is the basis of the Howe duality.
In fact, the Weil representation is the "minimal" representation of
M(2n,f).
I will define the notion of minimal (unitary) representation for
reductive groups over local fields, give explicit formulas for spherical
vectors for simply-laced groups, describe the space of smooth vectors and
the structure of the automorphic functionals.
Thursday (Feb 14), 4:30 p.m, room E 206.
David Kazhdan (Hebrew University).
Minimal representations of simply-laced reductive groups. II.
Abstract
For any local field F the Weil representation is a representation of
M(2n,f), the double cover of the group Sp(2n,F); this remarkable
representation is the basis of the Howe duality.
In fact, the Weil representation is the "minimal" representation of
M(2n,f).
I will define the notion of minimal (unitary) representation for
reductive groups over local fields, give explicit formulas for spherical
vectors for simply-laced groups, describe the space of smooth vectors and
the structure of the automorphic functionals.
Monday (Feb 18), 4:30 p.m, room E 206.
Alexander Efimov (Moscow).
Homotopy finiteness of DG categories from algebraic geometry.
[To understand the talk, it suffices to know standard facts about
triangulated and derived categories. In other words, don't be afraid of
words like "homotopy finiteness".]
Abstract
We will explain that for any separated scheme $X$ of finite type over a
field $k$ of characteristic zero, its derived category $D^b_{coh}(X)$
(considered as a DG category) is homotopically finitely presented over
$k$, confirming a conjecture of Kontsevich.
More precisely, we show that $D^b_{coh}(X)$ can be represented as a DG
quotient of some smooth and proper DG category $C$ by a subcategory
generated by a single object. This category $C$ has a semi-orthogonal
decomposition into derived categories of smooth and proper varieties. The
construction uses the categorical resolution of singularities of Kuznetsov
and Lunts, which in turn uses Hironaka Theorem.
A similar result holds also for the 2-periodic DG category $MF_{coh}(X,W)$
of coherent matrix factorizations on $X$ for any potential $W$.
Thursday (Feb 21), 4:30 p.m, room E 206.
Alexander Efimov (Moscow).
Homotopy finiteness of DG categories from algebraic geometry.II.
*******
Here are the references for the results mentioned in Efimov's first talk:
B. Toen, M. Vaquie, Moduli of objects in dg-categories, arXiv:math/0503269
Valery A. Lunts, Categorical resolution of singularities, arXiv:0905.4566
Raphael Rouquier, Dimensions of triangulated categories, arXiv:math/0310134
Alexei Bondal, Michel Van den Bergh, Generators and representability of
functors in commutative and noncommutative geometry, arXiv:math/0204218
Alexander Kuznetsov, Valery A. Lunts, Categorical resolutions of irrational
singularities, arXiv:1212.6170
M. Auslander, Representation dimension of Artin algebras, in Selected
works of Maurice Auslander. Part 1. American Mathematical Society,
Providence, RI, 1999.
No seminar on Monday (Feb 25).
******
On Thursday (Feb 28) there will be a
talk by Alexander Polishchuk (University of Oregon).
Title of his talk:
Matrix factorizations and cohomological field theories.
Abstract
This is joint work with Arkady Vaintrob.
I will explain how one can use DG categories of matrix factorizations to
construct a cohomological field theory associated with a quasihomogeneous
polynomial with isolated singularity at zero.
Thursday (Feb 28), 4:30 p.m, room E 206.
Alexander Polishchuk (University of Oregon).
Matrix factorizations and cohomological field theories.
Abstract
This is joint work with Arkady Vaintrob.
I will explain how one can use DG categories of matrix factorizations to
construct a cohomological field theory associated with a quasihomogeneous
polynomial with isolated singularity at zero.
Monday (March 4), 4:30 p.m, room E 206.
Richard Taylor (IAS). Galois representations for regular algebraic cusp
forms.
Abstract
I will start by reviewing what is expected, and what is known,
about the correspondence between algebraic l-adic representations of the
absolute Galois group of a number field and algebraic cuspidal
automorphic representations of GL(n) over that number field.
I will then discuss recent work with Harris, Lan and Thorne constructing
l-adic representations for regular algebraic cuspidal automorphic
representations of GL(n) over a CM field, without any self-duality
assumption on the automorphic representation. Without such an assumption
it is believed that these l-adic representations do not occur in the
cohomology of any Shimura variety, and we do not know how to construct
the corresponding motive (though we believe that a motive should exist).
Nonetheless we can construct the l-adic representations as an l-adic
limit of motivic l-adic representations.
No more meetings of the Geometric Langlands seminar this quarter.
The geometric Langlands seminar does not meet this week. On next Monday (April 8) Bhargav Bhatt will speak on Derived de Rham cohomology in characteristic 0. After that, on April 15 and 18 Ivan Losev will give lectures on categorifications of Kac-Moody algebras. (There are good reasons to expect his lectures to be understandable!)
Monday (April 8), 4:30 p.m, room E 206.
Bhargav Bhatt (IAS).
Derived de Rham cohomology in characteristic 0.
Abstract
Derived de Rham cohomology is a refinement of classical de Rham
cohomology of algebraic varieties that works better in the presence of
singularities; the difference, roughly, is the replacement of the
cotangent sheaf by the cotangent complex.
In my talk, I will first recall Illusie's definition of this
cohomology theory (both completed and non-completed variants). Then I
will explain why the completed variant computes algebraic de Rham
cohomology (and hence Betti cohomology) for arbitrary algebraic
varieties in characteristic 0; the case of local complete intersection
singularities is due to Illusie. As a corollary, one obtains a new
filtration on Betti cohomology refining the Hodge-Deligne filtration.
Another consequence that will be discussed is that the completed
Amitsur complex of a variety also calculates its algebraic de Rham
cohomology.
Monday (April 8), 4:30 p.m, room E 206.
Bhargav Bhatt (IAS).
Derived de Rham cohomology in characteristic 0.
Abstract
Derived de Rham cohomology is a refinement of classical de Rham
cohomology of algebraic varieties that works better in the presence of
singularities; the difference, roughly, is the replacement of the
cotangent sheaf by the cotangent complex.
In my talk, I will first recall Illusie's definition of this
cohomology theory (both completed and non-completed variants). Then I will
explain why the completed variant computes algebraic de Rham
cohomology (and hence Betti cohomology) for arbitrary algebraic
varieties in characteristic 0; the case of local complete intersection
singularities is due to Illusie. As a corollary, one obtains a new
filtration on Betti cohomology refining the Hodge-Deligne filtration.
Another consequence that will be discussed is that the completed
Amitsur complex of a variety also calculates its algebraic de Rham
cohomology.
No seminar on Thursday.
Next week Ivan Losev (Northeastern University) will speak on
Monday (April 15) and Thursday (April 15).
Title of Losev's lectures:
Introduction to categorical Kac-Moody actions.
Abstract
The goal of these lectures is to provide an elementary introduction to
categorical actions of Kac-Moody algebras from a representation theoretic
perspective.
In a naive way (which, of course, appeared first), a
categorical Kac-Moody action is a collection of
functors on a category that on the level of Grothendieck
groups give actions of the Chevalley generators of the Kac-Moody algebra.
Such functors were first observed in the representation theory of
symmetric
groups in positive characteristic and then for the BGG
category O of gl(n). Analyzing the examples, in 2004
Chuang and Rouquier gave a formal definition of a categorical
sl(2)-action. Later (about 2008) Rouquier and Khovanov-Lauda extended this
definition to arbitrary Kac-Moody Lie algebras.
Categorical Kac-Moody actions are very useful in
Representation theory and (potentially, at least) in
Knot theory. Their usefulness in Representation theory
is three-fold. First, they allow to obtain structural
results about the categories of interest (branching rules
for the symmetric groups obtained by Kleshchev,
or derived equivalences between different blocks
constructed by Chuang and Rouquier in order to
prove the Broue abelian defect conjecture).
Second, categories with Kac-Moody actions are often
uniquely determined by the "type of an action", sometimes
this gives character formulas. Third, the categorification business gives
rise to new
interesting classes of algebras that were not known before:
the KLR (Khovanov-Lauda-Rouquier) algebras.
Potential applications to Knot theory include categorical (hence
stronger) versions of quantum knot invariants, this area is
very much still in development.
I will start from scratch and try to keep the exposition elementary, in
particular I will only consider Kac-Moody algebras of type A, i.e., sl(n)
and \hat{sl(n)}. The most essential prerequisite is a good
understanding of the standard categorical language (e.g., functor
morphisms).
Familiarity with classical representation theoretic objects
such as affine Hecke algebras or BGG categories O is also useful
although these will be recalled.
A preliminary plan is as follows:
0) Introduction.
1) Examples: symmetric groups/type A Hecke algebras, BGG categories O. 2)
Formal definition of a categorical action.
3) More examples (time permitting): representations of GL.
4) Consequences of the definition: divided powers, categorifications of
reflections, categorical Serre relations, crystals.
5) Yet some more examples: cyclotomic Hecke algebras.
6) Structural results: minimal categorifications and their uniqueness,
filtrations, (time permitting) actions on highest weight categories,
tensor products.
Here are some important topics related to categorical Kac-Moody actions
that will not be discussed:
a) Categorical actions in other types and those of quantum groups. b)
Categorification of the algebras U(n),U(g), etc.
c) Connections to categorical knot invariants.
a) is described in reviews http://arxiv.org/abs/1301.5868
by Brundan and (a more advanced text) http://arxiv.org/abs/1112.3619 by
Rouquier. The latter also deals with b). A more basic review for b) is
http://arxiv.org/abs/1112.3619 by Lauda dealing with the sl_2 case and
also introducing diagrammatic calculus. I am not aware
of any reviews on c), a connection to Reshetikhin-Turaev
invariants was established in full generality by Webster
in http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.
Monday (April 15), 4:30 p.m, room E 206.
Ivan Losev (Northeastern University)
Introduction to categorical Kac-Moody actions.I
Abstract
The goal of this lecture and the one on April 18 is to provide an
elementary introduction to categorical actions of Kac-Moody algebras from
a representation theoretic perspective.
In a naive way (which, of course, appeared first), a categorical Kac-Moody
action is a collection of functors on a category that on the level of
Grothendieck groups give actions of the Chevalley generators of the
Kac-Moody algebra. Such functors were first observed in the
representation theory of symmetric groups in positive characteristic and
then for the BGG
category O of gl(n). Analyzing the examples, in 2004 Chuang and Rouquier
gave a formal definition of a categorical sl(2)-action. Later (about 2008)
Rouquier and Khovanov-Lauda extended this definition to arbitrary
Kac-Moody Lie algebras.
Categorical Kac-Moody actions are very useful in Representation theory and
(potentially, at least) in Knot theory. Their usefulness in Representation
theory is three-fold. First, they allow to obtain structural results about
the categories of interest (branching rules for the symmetric groups
obtained by Kleshchev, or derived equivalences between different blocks
constructed by Chuang and Rouquier in order to prove the Broue abelian
defect conjecture). Second, categories with Kac-Moody actions are often
uniquely determined by the "type of an action", sometimes this gives
character formulas. Third, the categorification business gives rise to new
interesting classes of algebras that were not known before: the KLR
(Khovanov-Lauda-Rouquier) algebras. Potential applications to Knot theory
include categorical (hence stronger) versions of quantum knot invariants,
this area is very much still in development.
I will start from scratch and try to keep the exposition elementary, in
particular I will only consider Kac-Moody algebras of type A, i.e., sl(n)
and \hat{sl(n)}. The most essential prerequisite is a good understanding
of the standard categorical language (e.g., functor morphisms).
Familiarity with classical representation theoretic objects such as affine
Hecke algebras or BGG categories O is also useful although these will be
recalled.
A preliminary plan is as follows:
0) Introduction.
1) Examples: symmetric groups/type A Hecke algebras, BGG categories O.
2) Formal definition of a categorical action.
3) More examples (time permitting): representations of GL.
4) Consequences of the definition: divided powers, categorifications of
reflections, categorical Serre relations, crystals.
5) Yet some more examples: cyclotomic Hecke algebras.
6) Structural results: minimal categorifications and their uniqueness,
filtrations, (time permitting) actions on highest weight categories,
tensor products.
Here are some important topics related to categorical Kac-Moody actions
that will not be discussed:
a) Categorical actions in other types and those of quantum groups.
b) Categorification of the algebras U(n),U(g), etc.
c) Connections to categorical knot invariants.
a) is described in reviews http://arxiv.org/abs/1301.5868
by Brundan and (a more advanced text)
http://arxiv.org/abs/1112.3619
by Rouquier. The latter also deals with b). A more basic review for b) is
http://arxiv.org/abs/1112.3619
by Lauda dealing with the sl_2 case and also introducing diagrammatic
calculus. I am not aware of any reviews on c), a connection to
Reshetikhin-Turaev invariants was established in full generality by
Webster in
http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.
Today, 4:30 p.m, room E 206.
Ivan Losev (Northeastern University)
Introduction to categorical Kac-Moody actions.I
Abstract
The goal of this lecture and the one on April 18 is to provide an
elementary introduction to categorical actions of Kac-Moody algebras from
a representation theoretic perspective.
In a naive way (which, of course, appeared first), a categorical Kac-Moody
action is a collection of functors on a category that on the level of
Grothendieck groups give actions of the Chevalley generators of the
Kac-Moody algebra. Such functors were first observed in the
representation theory of symmetric groups in positive characteristic and
then for the BGG
category O of gl(n). Analyzing the examples, in 2004 Chuang and Rouquier
gave a formal definition of a categorical sl(2)-action. Later (about 2008)
Rouquier and Khovanov-Lauda extended this definition to arbitrary
Kac-Moody Lie algebras.
Categorical Kac-Moody actions are very useful in Representation theory and
(potentially, at least) in Knot theory. Their usefulness in Representation
theory is three-fold. First, they allow to obtain structural results about
the categories of interest (branching rules for the symmetric groups
obtained by Kleshchev, or derived equivalences between different blocks
constructed by Chuang and Rouquier in order to prove the Broue abelian
defect conjecture). Second, categories with Kac-Moody actions are often
uniquely determined by the "type of an action", sometimes this gives
character formulas. Third, the categorification business gives rise to new
interesting classes of algebras that were not known before: the KLR
(Khovanov-Lauda-Rouquier) algebras. Potential applications to Knot theory
include categorical (hence stronger) versions of quantum knot invariants,
this area is very much still in development.
I will start from scratch and try to keep the exposition elementary, in
particular I will only consider Kac-Moody algebras of type A, i.e., sl(n)
and \hat{sl(n)}. The most essential prerequisite is a good understanding
of the standard categorical language (e.g., functor morphisms).
Familiarity with classical representation theoretic objects such as affine
Hecke algebras or BGG categories O is also useful although these will be
recalled.
A preliminary plan is as follows:
0) Introduction.
1) Examples: symmetric groups/type A Hecke algebras, BGG categories O. 2)
Formal definition of a categorical action.
3) More examples (time permitting): representations of GL.
4) Consequences of the definition: divided powers, categorifications of
reflections, categorical Serre relations, crystals.
5) Yet some more examples: cyclotomic Hecke algebras.
6) Structural results: minimal categorifications and their uniqueness,
filtrations, (time permitting) actions on highest weight categories,
tensor products.
Here are some important topics related to categorical Kac-Moody actions
that will not be discussed:
a) Categorical actions in other types and those of quantum groups. b)
Categorification of the algebras U(n),U(g), etc.
c) Connections to categorical knot invariants.
a) is described in reviews http://arxiv.org/abs/1301.5868
by Brundan and (a more advanced text)
http://arxiv.org/abs/1112.3619
by Rouquier. The latter also deals with b). A more basic review for b) is
http://arxiv.org/abs/1112.3619
by Lauda dealing with the sl_2 case and also introducing diagrammatic
calculus. I am not aware of any reviews on c), a connection to
Reshetikhin-Turaev invariants was established in full generality by
Webster in
http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.
Thursday (April 18), 4:30 p.m, room E 206. Ivan Losev. Introduction to categorical Kac-Moody actions.II.
No seminar on Monday (Apr 22) and Thursday (Apr 25).
The next meeting is on FRIDAY April 26 (4:30 p.m, room E 206).
(I do realize that Friday is not a very good day for a seminar, but
unfortunately, the speaker was unable to speak on another day.)
Friday (April 26), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT). Towards character sheaves for loop groups
Abstract
I will describe a joint project with D. Kazhdan and Y. Varshavsky whose
aim is to develop the theory of character sheaves for loop groups and
apply it to the theory of endoscopy for reductive $p$-adic groups. The
project started from an attempt to understand the relation of Lusztig's
classification of character sheaves (discussed in an earlier talk by the
speaker in this seminar) to local Langlands conjectures.
I will discuss results (to appear shortly) on a geometric proof of the
result by Kazhdan--Varshavsky and De Backer -- Reeder on stable
combinations of characters in a generic depth zero L-packet, and a proof
of the unramified case of the stable center conjecture. Time permitting, I
will describe a general approach to relating local geometric Langlands
duality to endoscopy.
Character sheaves on loop groups are also the subject of two recent papers
by Lusztig.
Losev's notes of his talks are here:
http://www.math.uchicago.edu/~mitya/langlands/LosevNotes.pdf
*******
Recall that the next meeting of the seminar is on FRIDAY:
Friday (April 26), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT). Towards character sheaves for loop groups
Abstract
I will describe a joint project with D. Kazhdan and Y. Varshavsky whose
aim is to develop the theory of character sheaves for loop groups and
apply it to the theory of endoscopy for reductive $p$-adic groups. The
project started from an attempt to understand the relation of Lusztig's
classification of character sheaves (discussed in an earlier talk by the
speaker in this seminar) to local Langlands conjectures.
I will discuss results (to appear shortly) on a geometric proof of the
result by Kazhdan--Varshavsky and De Backer -- Reeder on stable
combinations of characters in a generic depth zero L-packet, and a proof
of the unramified case of the stable center conjecture. Time permitting, I
will describe a general approach to relating local geometric Langlands
duality to endoscopy.
Character sheaves on loop groups are also the subject of two recent papers
by Lusztig.
No seminar on Monday (April 29). Dennis Gaitsgory will speak on Thursday (May 2) and Monday (May 6). The title of his talk will be announced soon.
Thursday (May 2), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard).
Eisenstein part of Geometric Langlands correspondence.I.
Abstract
Geometric Eisenstein series, Eis_!, is a functor
D-mod(Bun_T)->D-mod(Bun-G),
given by pull-push using the stack Bun_B is an intermediary.
Spectral Eisenstein series Eis_{spec} is a functor
QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}})
given by pull-push using the stack LocSys_{\check{B}} is an intermediary.
One of the expected key properties of the (still conjectural) Geometric
Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This
implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed
as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic
(here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric
Langlands for the torus, which is given by the Fourier-Mukai transform).
Vice versa, establishing the isomorphism of the above algebras of
endomorphisms is equivalent to establishing the Eisenstein part of the
Geometric Langlands equivalence. In these talks, we will indicate a
strategy toward the proof of this isomorphism. We will reduce the problem
from being global to one which is local (in particular, on the geometric
side, instead of Bun_G we will be dealing with the affine Grassmannian).
We will show that the local problem is equivalent to a factorizable
version of Bezrukavnikov's theory of Langlands duality for various
categories of D-modules on the affine Grassmannian.
Tomorrow (i.e., Thursday, May 2), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard).
Eisenstein part of Geometric Langlands correspondence.I.
Abstract
Geometric Eisenstein series, Eis_!, is a functor
D-mod(Bun_T)->D-mod(Bun-G),
given by pull-push using the stack Bun_B is an intermediary.
Spectral Eisenstein series Eis_{spec} is a functor
QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}})
given by pull-push using the stack LocSys_{\check{B}} is an intermediary.
One of the expected key properties of the (still conjectural) Geometric
Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This
implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed
as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic
(here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric
Langlands for the torus, which is given by the Fourier-Mukai transform).
Vice versa, establishing the isomorphism of the above algebras of
endomorphisms is equivalent to establishing the Eisenstein part of the
Geometric Langlands equivalence. In these talks, we will indicate a
strategy toward the proof of this isomorphism. We will reduce the problem
from being global to one which is local (in particular, on the geometric
side, instead of Bun_G we will be dealing with the affine Grassmannian).
We will show that the local problem is equivalent to a factorizable
version of Bezrukavnikov's theory of Langlands duality for various
categories of D-modules on the affine Grassmannian.
Monday (May 6), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard).
Eisenstein part of Geometric Langlands correspondence.II.
Abstract
Geometric Eisenstein series, Eis_!, is a functor
D-mod(Bun_T)->D-mod(Bun-G),
given by pull-push using the stack Bun_B is an intermediary.
Spectral Eisenstein series Eis_{spec} is a functor
QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}})
given by pull-push using the stack LocSys_{\check{B}} is an intermediary.
One of the expected key properties of the (still conjectural) Geometric
Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This
implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed
as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic
(here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric
Langlands for the torus, which is given by the Fourier-Mukai transform).
Vice versa, establishing the isomorphism of the above algebras of
endomorphisms is equivalent to establishing the Eisenstein part of the
Geometric Langlands equivalence. In these talks, we will indicate a
strategy toward the proof of this isomorphism. We will reduce the problem
from being global to one which is local (in particular, on the geometric
side, instead of Bun_G we will be dealing with the affine Grassmannian).
We will show that the local problem is equivalent to a factorizable
version of Bezrukavnikov's theory of Langlands duality for various
categories of D-modules on the affine Grassmannian.
Thursday (May 9), 4:30 p.m, room E 206.
Dustin Clausen (MIT). Arithmetic Duality in Algebraic K-theory.
Abstract
Let R be any commutative ring classically considered in
algebraic number theory (global field, local field, ring of integers...).
We will give a uniform definition of a ``compactly supported G-theory''
spectrum G_c(R) associated to R, supposed to be dual to the algebraic
K-theory K(R). Then, for every prime $\ell$ invertible in R, we will
construct a functorial $\ell$-adic pairing implementing this duality.
Finally, using work of Thomason connecting algebraic K-theory to Galois
theory, we will explain how these pairings allow to give a uniform
construction of the various Artin maps associated to such rings R, one by
which the Artin reciprocity law becomes tautological.
The crucial input is a simple homotopy-theoretic connection between tori,
real vector spaces, and spheres, which we hope to explain.
Monday (May 13), 4:30 p.m, room E 206.
Takako Fukaya. On non-commutative Iwasawa theory.
Abstract
Iwasawa theory studies a mysterious connection between algebraic
objects (ideal class groups, etc.) and analytic objects (p-adic Riemann
zeta functions etc.) in a p-adic way, considering certain p-adic infinite
towers of Galois extensions of number fields.
Historically, people first used infinite Galois extensions whose Galois
group is abelian. However, in recent years, non-commutative Iwasawa
theory, which considers infinite Galois extensions whose Galois group is
non-commutative has been developed. We will first review ``commutative
Iwasawa theory (usual Iwasawa theory)", then introduce the history of
non-commutative Iwasawa theory, and the results obtained recently.
No more meetings of the Langlands seminar this quarter.
As usual, the seminar will meet on Mondays and/or Thursdays in room E206
at 4:30 p.m.
We will begin with a series of talks by Beilinson on his recent work (the
title and abstract are below). In particular, he will give a proof of the
results of the article
http://arxiv.org/abs/1203.2776
(by Bloch, Esnault, and Kerz), which is more understandable and elementary
than the original one.
The first meeting is on October 10 (Thursday).
Alexander Beilinson. Relative continuous K-theory and cyclic homology.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology
of X) lies in the middle term of the Hodge filtration. A variant of the
deformational Hodge conjecture says that, up to torsion, this
condition is sufficient as well.
This conjecture remains a mystery, but in
a recent work "p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that
implies the Bloch-Esnault-Kerz theorem.
I will explain the background material, so no prior knowledge of the
subject is needed.
No seminar on Monday.
Thursday (Oct 10), 4:30 p.m, room E 206.
Alexander Beilinson will give his first talk on
Relative continuous K-theory and cyclic homology.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
I will explain the background material, so no prior knowledge of the
subject is needed.
Thursday (Oct 10), 4:30 p.m, room E 206.
Alexander Beilinson will give his first talk on
Relative continuous K-theory and cyclic homology.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
I will explain the background material, so no prior knowledge of the
subject is needed.
Thursday (Oct 10), 4:30 p.m, room E 206.
Alexander Beilinson will give his first talk on
Relative continuous K-theory and cyclic homology.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
I will explain the background material, so no prior knowledge of the
subject is needed.
Monday (Oct 14), 4:30 p.m, room E 206.
Alexander Beilinson.
Relative continuous K-theory and cyclic homology. II.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
No seminar this Thursday. Alexander Beilinson will continue on Monday (Oct 21).
Monday (Oct 21), 4:30 p.m, room E 206.
Alexander Beilinson.
Relative continuous K-theory and cyclic homology. III.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
Monday (Oct 21), 4:30 p.m, room E 206.
Alexander Beilinson.
Relative continuous K-theory and cyclic homology. III.
Abstract
Let X be a smooth proper scheme over the ring of p-adic integers Z_p.
Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed
fiber. How can one check if c comes from a class in K_0 (X)?
A necessary condition is that the Chern class ch(v) in the crystalline
cohomology of X/p (which is the same as de Rham cohomology of X) lies in
the middle term of the Hodge filtration. A variant of the deformational
Hodge conjecture says that, up to torsion, this condition is sufficient as
well.
This conjecture remains a mystery, but in a recent work
"p-adic deformation of algebraic cycle classes"
Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the
conjecture is valid if we replace K_0 (X) by the projective limit of
groups K_0 (X/p^n).
In this series of talks I will explain a p-adic version of Goodwillie's
theorem which identifies the relative continuous K-theory of a p-adic
associative algebra with its continuous cyclic homology, and that implies
the Bloch-Esnault-Kerz theorem.
No seminar on Thursday. Beilinson will continue on Monday (Oct 28).
No seminar on Monday (Oct 28);
Beilinson's talk has been CANCELED because quite unexpectedly, he has to
go to Moscow (his mother-in-law died).
****
Next Thursday (Oct 31) Steve Zelditch (NWU) will give his first talk on
Berezin-Toeplitz quantization.
Title of his talk:
Quantization and Toeplitz operators.
Abstract
One of the basic settings of geometric quantization is a Kahler manifold
(M, J, \omega), polarized by a Hermitian holomorphic line bundle $(L, h)
\to (M, \omega)$. The metric h induces inner products on the spaces
$H^0(M, L^k)$ of holomorphic sections of the k-th power of L. A basic
principle is that 1/k plays the role of Planck's constant, and one has
semi-classical asymptotics as k goes to infinity. The purpose of my first
lecture is to introduce the Szego kernels $\Pi_k$ in this context and to
explain why the semi-classical asymptotics exist. Toeplitz operators are
of the form $\Pi_k M_f \Pi_k$ where $M_f$ is multiplication by $f \in
C^{\infty}(M)$, and one gets a * product on the smooth functions by
composing operators. There is a more general formalism for almost complex
symplectic manifolds and in other settings.
Thursday (Oct 31), 4:30 p.m, room E 206.
Steve Zelditch (NWU) will give his first talk on
Quantization and Toeplitz operators.
Abstract
One of the basic settings of geometric quantization is a Kahler manifold
(M, J, \omega), polarized by a Hermitian holomorphic line bundle $(L, h)
\to (M, \omega)$. The metric h induces inner products on the spaces
$H^0(M, L^k)$ of holomorphic sections of the k-th power of L. A basic
principle is that 1/k plays the role of Planck's constant, and one has
semi-classical asymptotics as k goes to infinity. The purpose of my first
lecture is to introduce the Szego kernels $\Pi_k$ in this context and to
explain why the semi-classical asymptotics exist. Toeplitz operators are
of the form $\Pi_k M_f \Pi_k$ where $M_f$ is multiplication by $f \in
C^{\infty}(M)$, and one gets a * product on the smooth functions by
composing operators. There is a more general formalism for almost complex
symplectic manifolds and in other settings.
No seminar on Monday November 4. ***** The next meeting is on Thursday (Nov 7) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his second talk on Quantization and Toeplitz operators. Attached is a PDF file with Zelditch's notes of his first talk and the beginning of the second one. ***** Let me also tell you that on Monday November 11 Danny Calegari will give an introductory talk "Fundamental groups of Kahler manifolds".
Attachment:
Zelditch.pdf
Description: Adobe PDF document
The next meeting is on Thursday (Nov 7) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his second talk on Quantization and Toeplitz operators.
Monday (Nov 11), 4:30 p.m, room E 206.
Danny Calegari. Fundamental groups of Kahler manifolds (an introduction)
Abstract
I will try to explain some of what is known and not known about
fundamental groups of (closed) Kahler manifolds (hereafter "Kahler
groups"), especially concentrating on the constraints that arise for
geometric reasons, where "geometry" here is understood in the sense of a
geometric group theorist; so (for example), some of the tools I will
discuss include L^2 cohomology, Bieri-Neumann-Strebel invariants, and the
theory of harmonic maps to trees.
One reason to be interested in such groups is because nonsingular
projective varieties (over the complex numbers) are Kahler, so in
principle, constraints on Kahler groups (and their linear representations)
have implications for understanding local systems on projective varieties
(but I will not talk about this).
Most of what I want to discuss is classical, and has been well-known for
over 20 years, but I hope to discuss at least two interesting recent
developments:
(1) an elementary construction (due to Panov-Petrunin) to show that every
finitely presented group arises as the fundamental group of a compact
complex 3-fold (typically not projective!);
(2) a theorem of Delzant that a solvable Kahler group contains a nilpotent
group with finite index (the corresponding fact for fundamental groups of
nonsingular projective varieties is due to Arapura and Nori, and their
proof is very different).
This talk should be accessible to graduate students.
The next meeting is on Thursday (Nov 14) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his third talk on Quantization and Toeplitz operators. (Danny Calegary will finish his talk on \pi_1 of Kahler manifolds on Monday, Nov 18).
Attached is a file with Steve Zelditch's notes of his second and third lecture on "Quantization and Toeplitz operators" (The third lecture is today at 4:00 p.m.)
Attachment:
Zelditch lectures 2-3.pdf
Description: Adobe PDF document
Monday (Nov 18), 4:30 p.m, room E 206. Danny Calegari. Fundamental groups of Kahler manifolds. II
The next meeting is on Thursday (Nov 21) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his last talk on Quantization and Toeplitz operators. [On Monday (Nov 25) Kazuya Kato will speak on "Heights of motives".]
Monday (Nov 25), 4:30 p.m, room E 206.
Kazuya Kato. Heights of motives.
Abstract
The height of a rational number a/b (a, b integers which are coprime) is
defined as max(|a|, |b|). A rational number with small (resp. big) height
is a simple (resp. complicated) number. Though the notion height is so
naive, height has played fundamental roles in number theory.
There are important variants of this notion. In 1983, when Faltings proved
Mordell conjecture formulated in 1921, Faltings first proved Tate
conjecture for abelian varieties (it was also a great conjecture) by
defining heights of an abelian varieties, and then he deduced Mordell
conjecture from the latter conjecture.
In this talk, after I explain these things, I will explain that the
heights of abelian varieties by Faltings are generalized to heights of
motives. (Motive is thought of as a kind of generalization of abelian
variety.)
This generalization of height is related to open problems in number
theory. If we can prove finiteness of the number of motives of bounded
heights, we can prove important conjectures in number theory such as
general Tate conjecture and Mordell-Weil type conjectures in many cases.
Monday (Nov 25), 4:30 p.m, room E 206.
Kazuya Kato. Heights of motives.
Abstract
The height of a rational number a/b (a, b integers which are coprime) is
defined as max(|a|, |b|). A rational number with small (resp. big) height
is a simple (resp. complicated) number. Though the notion height is so
naive, height has played fundamental roles in number theory.
There are important variants of this notion. In 1983, when Faltings proved
Mordell conjecture formulated in 1921, Faltings first proved Tate
conjecture for abelian varieties (it was also a great conjecture) by
defining heights of an abelian varieties, and then he deduced Mordell
conjecture from the latter conjecture.
In this talk, after I explain these things, I will explain that the
heights of abelian varieties by Faltings are generalized to heights of
motives. (Motive is thought of as a kind of generalization of abelian
variety.)
This generalization of height is related to open problems in number
theory. If we can prove finiteness of the number of motives of bounded
heights, we can prove important conjectures in number theory such as
general Tate conjecture and Mordell-Weil type conjectures in many cases.
No more meetings of the seminar this quarter.
The first meeting of the seminar is on Jan 9.
Thursday (Jan 9), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin).
Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. I.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
an equivalence between two categories of a very different nature. One of
them is the derived category of D-modules on the moduli stack of principal
bundles on a curve. The other is a certain category of ind-coherent
sheaves on the moduli stack of local systems, which is a certain extension
of the derived category of quasi-coherent sheaves.
Of these two categories, the former is more familiar: its objects can be
viewed as geometric counterparts of automorphic forms. The category can be
studied using the Fourier transform, which yields a certain additional
structure on it. Roughly speaking, the category embeds into a larger
category (that of `Fourier coefficients'), which admits a natural
filtration indexed by conjugacy classes of parabolic subgroups.
In a joint project with D.Gaitsgory, we construct a similar structure on
the other side of the Langlands conjecture. Let LS(G) be the stack of
G-local systems, where G is a reductive group. For any parabolic subgroup
P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of
parabolic subgroups form an ordered set, and the corresponding stacks
LS(P) fit into a diagram over LS(G). Our main result is the embedding of
the category of ind-coherent sheaves on LS(G) into the category of
relative D-modules on this diagram. The result reduces to a purely
classical, but seemingly new, property of the topological (spherical)
Bruhat-Tits building of G.
In my talk, I plan to review the formalism of ind-coherent sheaves and the
role it plays in the categorical Langlands conjecture. I will show how
relative D-modules appear in the study of ind-coherent sheaves and how
the topological Bruhat-Tits building enters the picture.
Thursday (Jan 9), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin).
Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. I.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
an equivalence between two categories of a very different nature. One of
them is the derived category of D-modules on the moduli stack of principal
bundles on a curve. The other is a certain category of ind-coherent
sheaves on the moduli stack of local systems, which is a certain extension
of the derived category of quasi-coherent sheaves.
Of these two categories, the former is more familiar: its objects can be
viewed as geometric counterparts of automorphic forms. The category can be
studied using the Fourier transform, which yields a certain additional
structure on it. Roughly speaking, the category embeds into a larger
category (that of `Fourier coefficients'), which admits a natural
filtration indexed by conjugacy classes of parabolic subgroups.
In a joint project with D.Gaitsgory, we construct a similar structure on
the other side of the Langlands conjecture. Let LS(G) be the stack of
G-local systems, where G is a reductive group. For any parabolic subgroup
P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of
parabolic subgroups form an ordered set, and the corresponding stacks
LS(P) fit into a diagram over LS(G). Our main result is the embedding of
the category of ind-coherent sheaves on LS(G) into the category of
relative D-modules on this diagram. The result reduces to a purely
classical, but seemingly new, property of the topological (spherical)
Bruhat-Tits building of G.
In my talk, I plan to review the formalism of ind-coherent sheaves and the
role it plays in the categorical Langlands conjecture. I will show how
relative D-modules appear in the study of ind-coherent sheaves and how
the topological Bruhat-Tits building enters the picture.
Monday (Jan 13), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin).
Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. II.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
an equivalence between two categories of a very different nature. One of
them is the derived category of D-modules on the moduli stack of principal
bundles on a curve. The other is a certain category of ind-coherent
sheaves on the moduli stack of local systems, which is a certain extension
of the derived category of quasi-coherent sheaves.
Of these two categories, the former is more familiar: its objects can be
viewed as geometric counterparts of automorphic forms. The category can be
studied using the Fourier transform, which yields a certain additional
structure on it. Roughly speaking, the category embeds into a larger
category (that of `Fourier coefficients'), which admits a natural
filtration indexed by conjugacy classes of parabolic subgroups.
In a joint project with D.Gaitsgory, we construct a similar structure on
the other side of the Langlands conjecture. Let LS(G) be the stack of
G-local systems, where G is a reductive group. For any parabolic subgroup
P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of
parabolic subgroups form an ordered set, and the corresponding stacks
LS(P) fit into a diagram over LS(G). Our main result is the embedding of
the category of ind-coherent sheaves on LS(G) into the category of
relative D-modules on this diagram. The result reduces to a purely
classical, but seemingly new, property of the topological (spherical)
Bruhat-Tits building of G.
In my talk, I plan to review the formalism of ind-coherent sheaves and the
role it plays in the categorical Langlands conjecture. I will show how
relative D-modules appear in the study of ind-coherent sheaves and how
the topological Bruhat-Tits building enters the picture.
No seminar on Thursday (Jan 16).
On Monday (Jan 20) Dmitry Tamarkin (NWU) will give his first talk on
Microlocal theory of sheaves and its applications to symplectic topology.
Abstract
I will start with explaining some basics of the Kashiwara-Schapira
microlocal theory of sheaves on manifolds. This theory associates to any
sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of
$T^*M$ called the singular support of $S$. Using a 'conification' trick,
one can associate to any sheaf $F$ on $M\times R$ (satisfying certain
conditions) a (not-necessarily homogeneous) closed subset of $T^*M$,
Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity
outside of a compact), one constructs an endofunctor on an appropriate
full category of sheaves on $M\times R$, which transforms microsupports
in the obvious way. This allows one to solve some non-displaceability
questions in symplectic topology.
Monday (Jan 20), 4:30 p.m, room E 206.
Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications
to symplectic topology. I.
Abstract
I will start with explaining some basics of the Kashiwara-Schapira
microlocal theory of sheaves on manifolds. This theory associates to any
sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of
$T^*M$ called the singular support of $S$. Using a 'conification' trick,
one can associate to any sheaf $F$ on $M\times R$ (satisfying certain
conditions) a (not-necessarily homogeneous) closed subset of $T^*M$,
Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity
outside of a compact), one constructs an endofunctor on an appropriate
full category of sheaves on $M\times R$, which transforms microsupports
in the obvious way. This allows one to solve some non-displaceability
questions in symplectic topology.
No seminar on Thursday (Jan 23). Tamarkin will continue on Monday, January 27.
Monday (Jan 27), 4:30 p.m, room E 206.
Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications
to symplectic topology. II.
Abstract
I will start with explaining some basics of the Kashiwara-Schapira
microlocal theory of sheaves on manifolds. This theory associates to any
sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of
$T^*M$ called the singular support of $S$. Using a 'conification' trick,
one can associate to any sheaf $F$ on $M\times R$ (satisfying certain
conditions) a (not-necessarily homogeneous) closed subset of $T^*M$,
Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity
outside of a compact), one constructs an endofunctor on an appropriate
full category of sheaves on $M\times R$, which transforms microsupports
in the obvious way. This allows one to solve some non-displaceability
questions in symplectic topology.
Monday (Feb 3), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. III.
Monday (Feb 3), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. III.
No seminar on Thursday.
Nikita Nekrasov (Simons Center at Stony Brook)
will speak on Monday (Feb 10).
Title of his talk:
Geometric definition of the (q_1, q_2)-characters, and instanton fusion.
Abstract
I will give a geometric definition of a one-parametric deformation of
q-characters of the quantum affine and toroidal algebras, and discuss
their applications to the calculation of the instanton partition functions
of quiver gauge theories.
Monday (Feb 10), 4:30 p.m, room E 206.
Nikita Nekrasov (Simons Center for Geometry and Physics at Stony Brook).
Geometric definition of the (q_1, q_2)-characters, and instanton fusion.
Abstract
I will give a geometric definition of a one-parametric deformation of
q-characters of the quantum affine and toroidal algebras, and discuss
their applications to the calculation of the instanton partition functions
of quiver gauge theories.
Thursday (Feb 13), 4:30 p.m, room E 206.
Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem.
Abstract
For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}$,
$g : \mathbf{C}^{n+1} \to \mathbf{C}$ having an isolated critical point at
0 with value 0, the classical Thom-Sebastiani theorem describes the
vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and its monodromy) as a
tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where
$$(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n).$$
In this talk and in the subsequent one(s) I will discuss algebraic
variants and generalizations of this result over fields of any
characteristic, where the tensor product is replaced by a certain local
convolution product, as suggested by Deligne. The main theorem is a
Kunneth formula for $R\Psi$ in the framework of Deligne's theory of nearby
cycles over general bases, of which I will review the basics. At the end,
I will discuss questions logically independent of this, pertaining to the
comparison between convolution and tensor product in the tame case.
No seminar on Monday.
Luc Illusie will continue his talk on Thursday (Feb 20).
As mentioned in the yesterday talk, the key example of blow-up is
explained in Section 9 of Orgogozo's article available at
http://arxiv.org/abs/math/0507475
Oriented products are reviewed in Expos\'e XI from the book available at
http://www.math.polytechnique.fr/~orgogozo/travaux_de_Gabber/
Sabbah's example of "hidden blow-up" is contained in the following article:
Sabbah, Claude
Morphismes analytiques stratifi\'es sans \'eclatement et cycles
\'evanescents. C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), no. 1,
39-41.
Thursday (Feb 20), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. II.
No seminar on Monday. Luc Illusie will finish his talk on Thursday (Feb 27). The article by Laumon mentioned today is available here: http://www.numdam.org/numdam-bin/item?id=PMIHES_1987__65__131_0 The article by N.Katz with the proof of the Gabber-Katz theorem is here: http://www.numdam.org/item?id=AIF_1986__36_4_69_0 Relevant for Illusie's talk is the first part, in which Katz introduces a certain category of "special" finite etale coverings of the multiplicative group over a field of characteristic p; he shows that the category of such special coverings is equivalent to the category of all finite etale coverings of the punctured formal neighbourhood of infinity.
Thursday (Feb 27), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. III.
No seminar on Monday. Spencer Bloch will give Albert lectures on Friday, Monday, and Tuesday, see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml On Thursday (March 6) Dima Tamarkin will speak. Title of his talk: On Laplace transform Abstract: I will review the papers 'Integral kernels and Laplace transform' by Kashiwara-Schapira '97 and 'On Laplace transform' by d'Agnolo '2013. Both papers aim at describing Laplace transform images of various spaces of complex-analytic functions of tempered growth. In order to work with such spaces, a technique of ind-sheaves is used; the answers are given in terms of the Fourier-Sato transform and its non-homogeneous generalizations.
Today (March 6), 4:30 p.m, room E 206.
Dmitry Tamarkin (NWU). On Laplace transform.
Abstract
I will review the papers
'Integral kernels and Laplace transform' by Kashiwara-Schapira (1997) and
'On Laplace transform' by d'Agnolo (2013).
Both papers aim at describing Laplace transform images of various spaces
of complex-analytic functions of tempered growth. In order to work with
such spaces, a technique of ind-sheaves is used; the answers are given in
terms of the Fourier-Sato transform and its non-homogeneous
generalizations.
No more seminars this quarter. Tamarkin will explain d'Agnolo's work in spring.
No seminar this week. The first meeting is on April 7 (i.e., next Monday). Dmitry Tamarkin will speak on D'Agnolo's article "On the Laplace transform for tempered holomorphic functions".
Monday (April 7), 4:30 p.m, room E 206.
Dmitry Tamarkin (NWU). Laplace transform: non-homogeneous case.
Abstract
I am going to review d'Agnolo's paper "On the Laplace transform of
tempered holomorphic functions", see
http://arxiv.org/abs/1207.5278
His article focuses on defining the Laplace transform for certain spaces
of regular functions in several complex variables. This is a
generalization of the Kaschiwara-Schapira paper "Integral transforms with
exponential kernels and Laplace transform" (1997), which answers a similar
question for the spaces of tempered functions on homogeneous open subsets
(with respect to dilations of the complex space).
Here is one of the simplest corollaries of d'Agnolo's result. Let U be an
open pre-compact sub-analytic convex subset of a complex vector space V.
Let V' be the dual complex space and let h_A be the function on V'
defined as follows: h_A(y) is the infimum of Re(x,y) where x runs
through A. Let O^t(U) be the space of tempered holomorphic functions on
$U$. Let B^{p,q} be the space of (p,q)-forms on V' that grow (along with
the derivatives) no faster than a polynomial times e^{-h_A}. d'Agnolo's
construction provides an identification of O^t(U) with the quotient of
B^{n,n} by the delta bar image of B^{n,n-1}.
I am also planning to discuss a couple of other applications of
d'Agnolo's result.
No seminar on Thursday (April 10) and Monday (April 14). On April 17 (Thursday) Xinwen Zhu (NWU) will give his first talk on "Cycles on modular varieties via geometric Satake" (this is a more detailed version of the talk that he gave in June 2013 at the number theory seminar at UofC).
Thursday (April 17), 4:30 p.m, room E 206.
Xinwen Zhu (NWU). Cycles on modular varieties via geometric Satake. I.
Abstract
I will first describe certain conjectural Tate classes in the etale
cohomology of the special fibers of modular varieties (Shimura varieities
and the moduli space of Shtukas). According to the Tate conjecture, there
should exist corresponding algebraic cycles. Then I will use ideas from
geometric Satake to construct these conjectural cycles. This is based on a
joint work with Liang Xiao.
The construction consists of two parts. The first part is a
parametrization of the irreducible components of certain affine
Deligne-Lusztig varieties (and its mixed characteristic analogue). The
Mirkovic-Vilonen theory plays an important role here. By the Rapoport-Zink
uniformization, they provide the conjectural cycles. The second part is to
calculate the intersection matrix of these cycles (still work in
progress). Using the generalization of some recent ideas of V. Lafforgue,
we reduce this calculation to certain intersection numbers of cycles in
the affine Grassmannian, which again can be understood via geometric
Satake.
No seminar on Monday. Xinwen Zhu will give his next talk on Thursday April 24.
Thursday (April 24), 4:30 p.m, room E 206.
Xinwen Zhu (NWU). Cycles on modular varieties via geometric Satake. II.
Abstract
I will first describe certain conjectural Tate classes in the etale
cohomology of the special fibers of modular varieties (Shimura varieities
and the moduli space of Shtukas). According to the Tate conjecture, there
should exist corresponding algebraic cycles. Then I will use ideas from
geometric Satake to construct these conjectural cycles. This is based on a
joint work with Liang Xiao.
The construction consists of two parts. The first part is a
parametrization of the irreducible components of certain affine
Deligne-Lusztig varieties (and its mixed characteristic analogue). The
Mirkovic-Vilonen theory plays an important role here. By the Rapoport-Zink
uniformization, they provide the conjectural cycles. The second part is to
calculate the intersection matrix of these cycles (still work in
progress). Using the generalization of some recent ideas of V. Lafforgue,
we reduce this calculation to certain intersection numbers of cycles in
the affine Grassmannian, which again can be understood via geometric
Satake.
No seminar next week. Dima Arinkin will speak on the Monday after next week (i.e., on May 5).
Monday (May 5), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Cohomology of line bundles on
completely integrable systems.
(The talk is introductory in nature and will be accessible to
non-specialists).
Abstract
Let A be an abelian variety. The Fourier-Mukai transform gives an
equivalence between the derived category of quasicoherent sheaves on A and
the derived category of the dual abelian variety. The key step in the
construction of this equivalence is the computation of the cohomology of A
with coefficients in a topologically trivial line bundle.
In my talk, I will provide a generalization of this result to (algebraic)
completely integrable systems. Generically, an integrable system can be
viewed as a family of (Lagrangian) abelian varieties; however, special
fibers may be singular. We will show that the cohomology of fibers with
coefficients in topologically trivial line bundles are given by the same
formula (even if fibers are singular). The formula implies a `partial'
Fourier-Mukai transform for completely integrable systems.
No seminar on Thursday May 8 and Monday May 12. Zhiwei Yun (Stanford) will speak on Thursday May 15.
Thursday (May 15), 4:30 p.m, room E 206.
Zhiwei Yun (Stanford). Rigid automorphic representations and rigid local
systems.
Abstract
We define what it means for an automorphic representation of a reductive
group over a function field to be rigid. Under the Langlands
correspondence, we expect them to correspond to rigid local systems. In
general, rigid automorphic representations are easier to come up with than
rigid local systems, and the Langlands correspondence between the two can
be realized using techniques from the geometric Langlands program. Using
this observation we construct several new families of rigid local systems,
with applications to questions about motivic Galois groups and the inverse
Galois problem over Q.
Monday (May 19), 4:30 p.m, room E 206.
Alexander Goncharov (Yale). Hodge correlators and open string Hodge theory.
Abstract
Thanks to the work of Simpson, (which used results of Hitchin and
Donaldson) we have an action of the multiplicative group of C on
semisimple complex local systems on a compact Kahler manifold.
We define Hodge correlators for semisimple complex local systems on a
compact Kahler manifold, and show that they can be organized into an
"open string theory data".
Precisely, the category of semisimple local systems on a Kahler manifold
gives rise to a BV algebra. Given a family of Kahler manifolds over a base
B, these BV algebras form a variation (of pure twistor structures) on B.
The Hodge correlators are organized into a solution of the quantum Master
equation on B for this variation.
Here are two special cases of this construction when the base B is a point.
1. Consider the genus zero part of the Hodge correlators. We show that it
encodes a homotopy action of the twistor-Hodge Galois group by A-infinity
autoequivalences of the category of smooth complexes on X. It extends the
Simpson C^* action on semisimple local systems. It can be thought of as
the Hodge analog (for smooth complexes) of the Galois group action on the
etale site.
2. The simplest possible Hodge correlators on modular curves deliver
Rankin-Selberg integrals for the special values of L-functions of modular
forms at integral points, which, thanks to Beilinson, are known to be the
regulators of motivic zeta-elements.
We suggest that there is a similar open string structure on the category
of all holonomic D-modules.
No more meetings of the seminar this year. Note that this week there is a conference at NWU on "Representation Theory, Integrable Systems and Quantum Fields", see http://www.math.northwestern.edu/emphasisyear/
As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. The first meeting is on October 9 (Thursday). We will begin with talks by Gaitsgory (Oct 9 and possibly Oct 13) and by Bezrukavnikov (Oct 16 and possibly Oct 20).
October 9 (Thursday), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function
fields. I.
Abstract
This is a joint work with Jacob Lurie.
In the case of the function field of a curve X, the Tamagawa number
conjecture can be reformulated as the formula for the weighted sum of
isomorphism classes of G-bundles on X.
During the talk on Thursday we will show how this formula follows from the
Atiyah-Bott formula for the cohomology of the moduli space Bun_G of
G-bundles on X.
On Monday we will show how to deduce the Atiyah-Bott formula from another
local-to-global result, namely the so-called non-Abelian Poincare duality
(the latter says that Bun_G is uniformized by the affine Grassmannian of G
with contractible fibers). The deduction
{non-Abelian Poincare duality}-->{Atiyah-Bott formula}
will be based on performing Verdier duality on the Ran space of X. This
is a non-trivial procedure, and most of the talk will be devoted to
explanations of how to make it work.
Thursday (October 9), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function
fields. I.
Abstract
This is a joint work with Jacob Lurie.
In the case of the function field of a curve X, the Tamagawa number
conjecture can be reformulated as the formula for the weighted sum of
isomorphism classes of G-bundles on X.
During the talk on Thursday we will show how this formula follows from the
Atiyah-Bott formula for the cohomology of the moduli space Bun_G of
G-bundles on X.
On Monday we will show how to deduce the Atiyah-Bott formula from another
local-to-global result, namely the so-called non-Abelian Poincare duality
(the latter says that Bun_G is uniformized by the affine Grassmannian of G
with contractible fibers). The deduction
{non-Abelian Poincare duality}-->{Atiyah-Bott formula}
will be based on performing Verdier duality on the Ran space of X. This
is a non-trivial procedure, and most of the talk will be devoted to
explanations of how to make it work.
Monday (October 13), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function
fields. II.
Abstract
We will show how to deduce the Atiyah-Bott formula from another
local-to-global result, namely the so-called non-Abelian Poincare duality
(the latter says that Bun_G is uniformized by the affine Grassmannian of G
with contractible fibers). The deduction
{non-Abelian Poincare duality}-->{Atiyah-Bott formula}
will be based on performing Verdier duality on the Ran space of X. This
is a non-trivial procedure, and most of the talk will be devoted to
explanations of how to make it work.
Gaitsgory's article is attached.
The next meeting is on
Thursday (Oct 16), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT). Geometry of second adjointness for p-adic groups
Abstract
Basic operations in representation theory of reductive p-adic groups are
functors of parabolic induction and restriction (also known as Jacquet
functor). It is clear from the definitions that the induction functor is
right adjoint to the Jacquet functor. It was discovered by Casselman and
Bernstein in (or around) 1970's that the two functors satisfy also
another, less obvious adjointness. I will describe a joint work with
D.Kazhdan devoted to a geometric construction of this adjointness. We will
show that it comes from a map on spaces of functions which is formally
similar to (but is not known to be formally related to) nearby cycles for
D-modules.
Attachment:
Denis on Tamagawa.pdf
Description: Adobe PDF document
Bernstein's pre-print on second adjointness and his lectures on
representations of p-adic groups can be found at
http://www.math.uchicago.edu/~mitya/langlands.html
____________________________________________
No seminar on Monday.
____________________________________________
Thursday (Oct 23), 4:30 p.m, room E 206.
Amnon Yekutiel (Ben Gurion University). Local Beilinson-Tate Operators.
Abstract
In 1968 Tate introduced a new approach to residues on algebraic curves,
based on a certain ring of operators that acts on the completion at a
point of the function field of the curve. This approach was generalized to
higher dimensional algebraic varieties by Beilinson in 1980. However
Beilinson's paper had very few details, and his operator-theoretic
construction remained cryptic for many years. Currently there is a renewed
interest in the Beilinson-Tate approach to residues in higher dimensions
(by Braunling, Wolfson and others). This current work also involves
n-dimensional Tate spaces and is related to chiral algebras.
In this talk I will discuss my recent paper arXiv:1406.6502, with same
title as the talk. I introduce a variant of Beilinson's operator-theoretic
construction. I consider an n-dimensional topological local field (TLF) K,
and define a ring of operators E(K) that acts on K, which I call the ring
of local Beilinson-Tate operators. My definition is of an analytic nature
(as opposed to the original geometric definition of Beilinson). I study
various properties of the ring E(K).
In particular I show that E(K) has an n-dimensional cubical decomposition,
and this gives rise to a residue functional in the style of
Beilinson-Tate. I conjecture that this residue functional coincides with
the residue functional that I had constructed in 1992 (itself an improved
version of the residue functional of Parshin-Lomadze).
Another conjecture is that when the TLF K arises as the Beilinson
completion of an algebraic variety along a maximal chain of points, then
the ring of operators E(K) that I construct, with its cubical
decomposition (the depends only on the TLF structure of K), coincides with
the cubically decomposed ring of operators that Beilinson constructed in
his original paper (and depends on the geometric input).
In the talk I will recall the necessary background material on
semi-topological rings, high dimensional TLFs, the TLF residue functional
and the Beilinson completion operation (all taken from Asterisque 208).
Thursday (Oct 23), 4:30 p.m, room E 206.
Amnon Yekutieli (Ben Gurion University). Local Beilinson-Tate Operators.
Abstract
In 1968 Tate introduced a new approach to residues on algebraic curves,
based on a certain ring of operators that acts on the completion at a
point of the function field of the curve. This approach was generalized to
higher dimensional algebraic varieties by Beilinson in 1980. However
Beilinson's paper had very few details, and his operator-theoretic
construction remained cryptic for many years. Currently there is a renewed
interest in the Beilinson-Tate approach to residues in higher dimensions
(by Braunling, Wolfson and others). This current work also involves
n-dimensional Tate spaces and is related to chiral algebras.
In this talk I will discuss my recent paper arXiv:1406.6502, with same
title as the talk. I introduce a variant of Beilinson's operator-theoretic
construction. I consider an n-dimensional topological local field (TLF) K,
and define a ring of operators E(K) that acts on K, which I call the ring
of local Beilinson-Tate operators. My definition is of an analytic nature
(as opposed to the original geometric definition of Beilinson). I study
various properties of the ring E(K).
In particular I show that E(K) has an n-dimensional cubical decomposition,
and this gives rise to a residue functional in the style of
Beilinson-Tate. I conjecture that this residue functional coincides with
the residue functional that I had constructed in 1992 (itself an improved
version of the residue functional of Parshin-Lomadze).
Another conjecture is that when the TLF K arises as the Beilinson
completion of an algebraic variety along a maximal chain of points, then
the ring of operators E(K) that I construct, with its cubical
decomposition (the depends only on the TLF structure of K), coincides with
the cubically decomposed ring of operators that Beilinson constructed in
his original paper (and depends on the geometric input).
In the talk I will recall the necessary background material on
semi-topological rings, high dimensional TLFs, the TLF residue functional
and the Beilinson completion operation (all taken from Asterisque 208).
Monday (Oct 27), 4:30 p.m, room E 206.
Adam Gal (Tel Aviv University). Self-adjoint Hopf categories and
Heisenberg categorification.
Abstract
We use the language of higher category theory to define what we call a
"symmetric self-adjoint Hopf" (SSH) structure on a semisimple abelian
category, which is a categorical analog of Zelevinsky's positive
self-adjoint Hopf algebras. As a first result, we obtain a categorical
analog of the Heisenberg double and its Fock space action, which is
constructed in a canonical way from the SSH structure.
No seminar on Thursday.
______________________________
Monday (Nov 3), 4:30 p.m, room E 206.
Francis Brown (IHES). Periods, iterated integrals and modular forms.
Abstract
It is conjectured that there should be a Galois theory of certain
transcendental numbers called periods. Using this as motivation, I will
explain how the notion of motivic periods gives a setting in which this
can be made to work. The goal is then to use geometry to compute the
Galois action on interesting families of (motivic) periods.
I will begin with the projective line minus three points, whose periods
are multiple zeta values, and try to work up to the upper half plane
modulo SL_2(Z), whose periods correspond to multiple versions of L-values
of modular forms.
Monday (Nov 3), 4:30 p.m, room E 206.
Francis Brown (IHES). Periods, iterated integrals and modular forms.
Abstract
It is conjectured that there should be a Galois theory of certain
transcendental numbers called periods. Using this as motivation, I will
explain how the notion of motivic periods gives a setting in which this
can be made to work. The goal is then to use geometry to compute the
Galois action on interesting families of (motivic) periods.
I will begin with the projective line minus three points, whose periods
are multiple zeta values, and try to work up to the upper half plane
modulo SL_2(Z), whose periods correspond to multiple versions of L-values
of modular forms.
Thursday (Nov 6), 4:30 p.m, room E 206. Francis Brown will continue on Thursday Nov 6 (4:30 p.m, room E 206).
Monday (Nov 10), 4:30 p.m, room E 206.
Sam Raskin (MIT). Chiral principal series categories. I.
Abstract
We will discuss geometric Langlands duality for unramified principal
series categories. This generalizes (in a roundabout way) some previous
work in local geometric Langlands to the setting where points in a curve
are allowed to move and collide. Using this local theory, we obtain
applications to the global geometric program, settling a conjecture of
Gaitsgory in the theory of geometric Eisenstein series.
Thursday (Nov 13), 4:30 p.m, room E 206. Sam Raskin (MIT). Chiral principal series categories. II.
Attached is Sam Raskin's write-up on "D-modules in infinite type", which could help you understand his yesterday talk. As I said, Sam will give his second talk on Thursday (Nov 13), 4:30 p.m, room E 206.
Attachment:
D-modules in infinite type.pdf
Description: Adobe PDF document
1. Sam Raskin's notes of his talks are attached.
2. No seminar next week.
3. Afterward, Keerthi Madapusi Pera will give several talks. I asked him
to us some "fairy tales" about Shimura varieties which appear as quotients
of the symmetric space SO(2,n)/{SO(2)\times SO(n)}. (Here "fairy tale"
means "an understandable talk for non-experts about something truly
mysterious mathematical objects".)
Keerthi will speak on (some of) the following dates: Nov 24, Dec 1, Dec 4.
The date of his first talk and the title&abstract of his series of talks
will be announced later.
Attachment:
Sam Raskin's notes.pdf
Description: Adobe PDF document
Monday (Nov 24), 4:30 p.m, room E 206.
Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical
models. I.
Abstract
The protagonists of the talk are arithmetic quotients of certain real
semi-algebraic Grassmannians associated with quadratic spaces of signature
(n,2). They are natural generalizations of the modular curves: the upper
half plane can be seen as a real Grassmannian of signature (1,2). In
certain cases, these spaces are also closely related to the moduli spaces
for K3 surfaces.
Quite miraculously, it turns out that these spaces are quasi-projective
algebraic varieties defined over the rational numbers, and even the
integers. One reason this is surprising is that they are not known to be
the solution to any natural moduli problem. However, due to the work of
many people, beginning with Deligne, we can say quite a bit about them by
using the 'motivic' properties of cohomological cycles on abelian
varieties.
This talk will mainly be a leisurely explication of this last sentence.
Monday (Dec 1), 4:30 p.m, room E 206. Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical models. II.
Monday (Dec 1), 4:30 p.m, room E 206. Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical models. II.
No more meetings of the seminar this quarter.
The first meeting is on Thursday (Jan 8), 4:30 p.m, room E 206.
Jacob Lurie will give two unrelated talks on
Thursday (Jan 8) and Monday (Jan 12).
The titles and abstracts are below.
******
Thursday (Jan 8), 4:30 p.m, room E 206.
Jacob Lurie (Harvard). Roots of Unity in Stable Homotopy Theory
Abstract
In classical algebraic geometry, there is often a stark difference between
the behavior of fields of characteristic zero (such as the complex numbers)
and fields of characteristic p (such as finite fields). For example, the
equation x^p = 1 has p distinct solutions over the field of complex
numbers, but only one solution over any field of characteristic p. In this
talk, I'll introduce the subject of K(n)-local stable homotopy theory,
which in some sense interpolates between characteristic zero and
characteristic p, and describe the curious behavior of roots of unity in
this intermediate regime.
******
Monday (Jan 12), 4:30 p.m, room E 206.
Jacob Lurie (Harvard). Rotation Invariance in Algebraic K-Theory
Abstract
For any triangulated category C, one can introduce an abelian group K_0(C)
which is freely generated by symbols [X]
where X is an object of C, subject to the relation [X] = [X'] + [X'']
whenever there is a distinguished triangle X' -> X -> X''.
This relation immediately implies that the double suspension map from C to
itself induces the identity map from K_0(C) to K_0(C).
In this talk, I will describe a "delooping" of this observation, which
asserts that the formation of algebraic K-theory is equivariant with
respect to a certain action of the circle group U(1).
Thursday (Jan 8), 4:30 p.m, room E 206.
Jacob Lurie (Harvard). Roots of Unity in Stable Homotopy Theory
Abstract
In classical algebraic geometry, there is often a stark difference between
the behavior of fields of characteristic zero (such as the complex
numbers) and fields of characteristic p (such as finite fields). For
example, the equation x^p = 1 has p distinct solutions over the field of
complex numbers, but only one solution over any field of characteristic p.
In this talk, I'll introduce the subject of K(n)-local stable homotopy
theory, which in some sense interpolates between characteristic zero and
characteristic p, and describe the curious behavior of roots of unity in
this intermediate regime.
Monday (Jan 12), 4:30 p.m, room E 206.
Jacob Lurie (Harvard). Rotation Invariance in Algebraic K-Theory
Abstract
For any triangulated category C, one can introduce an abelian group K_0(C)
which is freely generated by symbols [X]
where X is an object of C, subject to the relation [X] = [X'] + [X'']
whenever there is a distinguished triangle X' -> X -> X''.
This relation immediately implies that the double suspension map from C to
itself induces the identity map from K_0(C) to K_0(C).
In this talk, I will describe a "delooping" of this observation, which
asserts that the formation of algebraic K-theory is equivariant with
respect to a certain action of the circle group U(1).
No seminar on Thursday (Jan 15) and Monday (Jan 19). On Thursday next week (i.e., on Jan 22) there will be a talk by Carlos Simpson.
Thursday (Jan 22), 4:30 p.m, room E 206.
Carlos Simpson (University of Nice Sophia Antipolis). Constructing
two-dimensional buildings.
Abstract
This reports on work in progress with Katzarkov, Noll and Pandit. We would
like to generalize the leaf-space tree of a quadratic differential, to
spectral curves for higher-rank Higgs bundles. Our current work concerns
$SL_3$. In this case the corresponding buildings have dimension two. Given
a spectral curve corresponding to multivalued differential $(\phi _1,\phi
_2,\phi _3)$ we propose a construction by a successive series of cut and
paste steps, of a universal pre-building. The distance function in this
pre-building calculates the exponent for any WKB problem with
limiting spectral curve $\phi$. The construction is conditioned on
non-existence of BPS states in the Gaiotto-Moore-Neitzke spectral network.
No seminar on Monday.
*********
Thursday (Jan 29), 4:30 p.m, room E 206.
Simon Schieder (Harvard). The Drinfeld-Lafforgue-Vinberg degeneration of
the stack of G-bundles.
Abstract
We study the singularities of the Drinfeld-Lafforgue-Vinberg
compactification of the moduli stack of G-bundles on a smooth projective
curve for a reductive group G. The definition of this compactification is
due to Drinfeld and relies on the Vinberg semigroup of G. We will mostly
focus on the case G=SL_2; in this case the compactification can
alternatively be viewed as a canonical one-parameter degeneration of the
moduli stack of SL_2-bundles. We then study the singularities of this
one-parameter degeneration via the associated nearby cycles construction.
Time permitting, we might sketch a generalization to the case of an
arbitrary reductive group G and the relation to Langlands duality.
Thursday (Jan 29), 4:30 p.m, room E 206.
Simon Schieder (Harvard). The Drinfeld-Lafforgue-Vinberg degeneration of
the stack of G-bundles.
Abstract
We study the singularities of the Drinfeld-Lafforgue-Vinberg
compactification of the moduli stack of G-bundles on a smooth projective
curve for a reductive group G. The definition of this compactification is
due to Drinfeld and relies on the Vinberg semigroup of G. We will mostly
focus on the case G=SL_2; in this case the compactification can
alternatively be viewed as a canonical one-parameter degeneration of the
moduli stack of SL_2-bundles. We then study the singularities of this
one-parameter degeneration via the associated nearby cycles construction.
Time permitting, we might sketch a generalization to the case of an
arbitrary reductive group G and the relation to Langlands duality.
No seminar on Monday.
*********
Thursday (Feb 5), 4:30 p.m, room E 206.
Ivan Mirkovic (Univ. of Massachusetts, Amherst): Loop Grassmannians from
the point of view of local spaces.
Abstract
The loop Grassmannians of reductive groups will be reconstructed in the
setting of “local spaces” over a curve. The structure of a local space is
a version of the fundamental structure of a factorization space introduced
and developed by Beilinson and Drinfeld. The weakening of the requirements
formalizes some well known examples of “almost factorization spaces'' .
The change of emphases leads to new constructions.
The main example will be generalizations of loop Grassmannians
corresponding to quadratic forms Q on based lattices. The quadratic form
corresponding to the loop Grassmannian of a simply connected group G is
essentially the "basic level" of G.
Thursday (Feb 5), 4:30 p.m, room E 206.
Ivan Mirkovic (Univ. of Massachusetts, Amherst): Loop Grassmannians from
the point of view of local spaces.
Abstract
The loop Grassmannians of reductive groups will be reconstructed in the
setting of “local spaces” over a curve. The structure of a local space is
a version of the fundamental structure of a factorization space introduced
and developed by Beilinson and Drinfeld. The weakening of the requirements
formalizes some well known examples of “almost factorization spaces'' .
The change of emphases leads to new constructions.
The main example will be generalizations of loop Grassmannians
corresponding to quadratic forms Q on based lattices. The quadratic form
corresponding to the loop Grassmannian of a simply connected group G is
essentially the "basic level" of G.
Ivan Mirkovic will continue his talk on Monday (Feb 9), 4:30 p.m, room E 206. The talk will recall the so called zastava spaces which appear in several places in the geometric representation theory.. The goal is to make them accessible by comparing different points of view and emphasizing the examples and the visual intuition given by the corresponding polytopes. (Ivan's first talk on generalizing loop Grassmannians tried to introduce the characters of the story and one these was the zastava. However, the two talks are independent of each other.) PS. Ivan's web page http://people.math.umass.edu/~mirkovic/ now contains a section "NOTES on Loop Grassmannians, Zastava Spaces.Semiinfinite Grassmannians". Some of these may be helpful (but not necessary). 1. The various definitions of Zastava spaces are compared in - Zastava Spaces <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/BC.ZastavaSpaces.pdf> 2. The Zastva spaces are the Beilinson-Drinfeld deformations of intersections of the opposite semiinfinite orbits in Loop Grassmannians. The semiinfinite orbits and their intersections contain the information on the negative part of the enveloping algebra of the Langlands dual of our reductive group. This is not essential for understanding the zastava spaces but is a useful part of the landscape. - Loop Grassmannian construction of the negative part of the Enveloping Algebra for the Langlands dual group. <http://www.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/ALGEBRAS/LoopGrassmannianConstruction.of.NegativeEnvelopingAlgebra.pdf> 3. The zastava spaces appeared in the paper Smiinfinite Flags I and 2 with Finkelberg, Feigin, Kuznetsov. These papers also contain mujch more -- the computation of the intersection cohomology of zastava spaces, a construction of a skeleton of the semiinfinite Grassmannian, a construction of the enveloping algebra of the Langlands dual group etc. This is surveyed in > - > <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/BC.ZastavaSpaces.pdf> - Notes on the papers Semiinfinite Flags I and II. > <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/SemiinfiniteFlagsPapers.Notes.pdf>
Ivan Mirkovic will continue his talk on Zastava spaces on Thursday (Feb 12), 4:30 p.m, room E 206.
Ivan Mirkovic will continue his talk on Zastava spaces on Monday (Feb 16), 4:30 p.m, room E 206.
No seminar on Thursday.
On Monday (Feb 23) Ngo Bao Chau will begin to speak on his recent work.
Title: Local unramified L-factor and singularity in a reductive monoid.
Abstract
We are interested in the (unramified) test function on a reductive group
over a non-archimedean local field which gives rise to a local
(unramified) factor of Langlands' automorphic L-function. Langlands'
automorphic L-function depends on an algebraic representation of the dual
group. In the case of the standard representation of GL(n), this test
function is essentially the characteristic function of the space of
matrices with integral coefficients, according to Godement-Jacquet. For a
general representation, the test function is related to singularity of a
reductive monoid which was constructed by Braverman and Kazhdan.
Monday (Feb 23), 4:30 p.m, room E 206.
Ngo Bao Chau. Local unramified L-factor and singularity in a reductive
monoid
Abstract
We are interested in the (unramified) test function on a reductive group
over a non-archimedean local field which gives rise to a local
(unramified) factor of Langlands' automorphic L-function. Langlands'
automorphic L-function depends on an algebraic representation of the dual
group. In the case of the standard representation of GL(n), this test
function is essentially the characteristic function of the space of
matrices with integral coefficients, according to Godement-Jacquet. For a
general representation, the test function is related to singularity of a
reductive monoid which was constructed by Braverman and Kazhdan.
Monday (March 2), 4:30 p.m, room E 206.
Ngo Bao Chau. Local unramified L-factor and singularity in a reductive
monoid. II.
Abstract
We are interested in the (unramified) test function on a reductive group
over a non-archimedean local field which gives rise to a local
(unramified) factor of Langlands' automorphic L-function. Langlands'
automorphic L-function depends on an algebraic representation of the dual
group. In the case of the standard representation of GL(n), this test
function is essentially the characteristic function of the space of
matrices with integral coefficients, according to Godement-Jacquet. For a
general representation, the test function is related to singularity of a
reductive monoid which was constructed by Braverman and Kazhdan.
Monday (March 2), 4:30 p.m, room E 206.
Ngo Bao Chau. Local unramified L-factor and singularity in a reductive
monoid. II.
Abstract
We are interested in the (unramified) test function on a reductive group
over a non-archimedean local field which gives rise to a local
(unramified) factor of Langlands' automorphic L-function. Langlands'
automorphic L-function depends on an algebraic representation of the dual
group. In the case of the standard representation of GL(n), this test
function is essentially the characteristic function of the space of
matrices with integral coefficients, according to Godement-Jacquet. For a
general representation, the test function is related to singularity of a
reductive monoid which was constructed by Braverman and Kazhdan.
No more meetings of the seminar this quarter.
**********
Jochen Heinloth from University of Essen will give a series of lectures
starting on this Friday, March 6, 2-3PM, room E203. The topic of his
lectures will be:
An introduction to the P=W conjecture and related conjectures of Hausel.
Abstract
I will try to explain the work of Hausel and Rodriguez-Villegas and de
Cataldo-Hausel-Migliorini resulting in a series of conjectures
on the global geometry of moduli spaces of Higgs bundles.
The starting point will be the different algebraic structures on the
manifold underlying the moduli space of Higgs bundles on a curve. Hausel
and Rodriguez-Villegas managed to do point counting arguments in one of
the complex structures (the character variety) and using this, they found
interesting properties of the cohomology that are reminiscent of
properties of intersection cohomology. This finally led de
Cataldo-Hausel-Migliorini to propose the P=W conjecture which they could
prove in some cases.
No meetings of the seminar during the first week of the quarter.
First meeting: Monday April 6 (4:30 p.m, room E 206).
On April 6 Nick Rozenblyum will give the first talk in a series devoted to
his joint work with David Ayala and John Francis.
Title: Higher categories and manifold topology. I.
Abstract
Over the past few decades, there has been a fruitful interplay between
manifold topology and (higher) category theory. I will give an overview of
some of these connections, and discuss joint work with David Ayala and
John Francis, which describes higher categories in terms of the topology
of stratified manifolds. This approach provides a precise dictionary
between manifold topology and higher category theory, and makes numerous
connections between the two manifest.
Monday (April 6), 4:30 p.m, room E 206.
Nick Rozenblyum. Higher categories and manifold topology.
This is the first talk in a series. Most probably, Nick will also speak on
Thursday April 9. The next talks will be given by David Ayala, see
http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands
Abstract
Over the past few decades, there has been a fruitful interplay between
manifold topology and (higher) category theory. I will give an overview of
some of these connections, and discuss joint work with David Ayala and
John Francis, which describes higher categories in terms of the topology
of stratified manifolds. This approach provides a precise dictionary
between manifold topology and higher category theory, and makes numerous
connections between the two manifest.
Thursday (Apr 9), 4:30 p.m, room E 206. Nick Rozenblyum. Higher categories and manifold topology.II.
No seminar on Monday (Apr 13).
On Thursday Apr 16 David Ayala will give his first talk.
Title: Higher categories are sheaves on manifolds.
Abstract
This project is an effort to merge higher algebra/category theory and
differential topology. As an outcome, information flows in both
directions: coherent constructions of manifold and embedding invariants
from higher algebraic/categorical data, such as that of a representation
of a quantum group lending to knot invariants; deformations of higher
algebraic/categorical parameters indexed by manifolds, such as Hochschild
(co)homology.
The talks will be framed by one main result, and a couple formal
applications thereof. The main construction is factorization homology
with coefficients in higher categories. The body of the talks will focus
on essential aspects of our definitions that facilitate the coherent
cancelations that support our main result.
This is joint work with John Francis and Nick Rozenblyum.
Thursday (Apr 16), 4:30 p.m, room E 206.
David Ayala (Montana State University).
Higher categories are sheaves on manifolds. I.
Abstract
This project is an effort to merge higher algebra/category theory and
differential topology. As an outcome, information flows in both
directions: coherent constructions of manifold and embedding invariants
from higher algebraic/categorical data, such as that of a representation
of a quantum group lending to knot invariants; deformations of higher
algebraic/categorical parameters indexed by manifolds, such as Hochschild
(co)homology.
The talks will be framed by one main result, and a couple formal
applications thereof. The main construction is factorization homology
with coefficients in higher categories. The body of the talks will focus
on essential aspects of our definitions that facilitate the coherent
cancelations that support our main result.
This is joint work with John Francis and Nick Rozenblyum.
Monday (Apr 20), 4:30 p.m, room E 206. David Ayala. Higher categories are sheaves on manifolds. II.
Thursday (April 23), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan). Integral p-adic Hodge theory.
Abstract
Let X be a smooth and proper scheme over the ring of integers in a p-adic
field. Classical p-adic Hodge theory relates the etale and de Rham
cohomologies of X: the theories are naturally identified after extending
scalars to a suitable ring of periods constructed by Fontaine. This
isomorphism is compatible with Galois actions, and thus plays a crucial
role in our understanding of Galois representations. This identification,
however, neglects all torsion phenomena as the period ring is a Q-algebra.
In my talk, I will briefly recall this rational story, and then describe a
new "comparison" between these two cohomology theories that works
integrally: we will realize the de Rham cohomology of X as a
specialization of the etale cohomology, integrally, over a 2-parameter
base. As an application, we deduce the optimal result relating torsion in
the two theories: the torsion in de Rham cohomology is an upper bound for
the torsion in etale cohomology (and the inequality can be strict). This
inequality can be used to explain some of the "pathologies" in de Rham
cohomology in characteristic p.
(Based on joint work in progress with Morrow and Scholze.)
No seminar next week.
1. No seminar this week. 2. Joseph Bernstein will be visiting our department starting from Wednesday May 6. He will give a seminar talk on Monday May 11 and at least one talk after that (on May 18 and maybe May 21). Yiannis Sakellaridis will be visiting us on May 13-16; he will speak on May 14. The titles and abstracts can be found at http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands 3. A mysterious theorem is formulated on p.2 of the article http://arxiv.org/pdf/math/0701615v3.pdf by Kumar, Lusztig, and Dipendra Prasad. As explained there, the theorem is proved in Jantzen's Ph.D. thesis (1973). An equivalent formulation is given in the Corollary (also on p.2). DOES ANYBODY KNOW A CONCEPTUAL EXPLANATION of the result? (E.g., is there any categorical statement behind it?) The result is as follows. Let G be a connected simply connected almost-simple group equipped with a pinning (English translation of Bourbaki's "epinglage"). Let \sigma be a nontrivial automorphism of the Dynkin diagram, then \sigma acts on G. Let G^\sigma denote the subgroup of fixed points. Let G_\sigma denote the simply connected group whose root system is dual to that of G^\sigma . The Corollary on p.2 says that the \sigma-characters of G are equal to the characters of G_\sigma . (This is mysterious because there is no apparent relation between G and G_\sigma and also because passing to the dual root system is just a formal combinatorial operation).
Monday (May 11), 4:30 p.m, room E 206.
Joseph Bernstein (Tel Aviv University). Stacks in Representation Theory.
(What is a representation of an algebraic group?)
Abstract
I will discuss a new approach to representation theory of algebraic groups.
In the usual approach one starts with an algebraic group G over some local
(or finite) field F, considers the group G(F) of its F-points as a
topological group and studies some category Rep (G(F)) of continuous
representations of the group G(F).
I will argue that more correct objects to study are some kind of sheaves
on the stack BG corresponding to the group G.
I will show that this point of view naturally requires to change the
formulation of some basic problems in Representation Theory. In
particular, this approach might explain the appearance of representations
of all pure forms of a group G in Vogan's formulation of Langlands'
correspondence.
Thursday (May 14), 4:30 p.m, room E 206.
Yiannis Sakellaridis (Rutgers University). Spectral decomposition on
homogeneous spaces.
Abstract
I will present results from my joint work with Venkatesh, Delorme and
Harinck on harmonic analysis on homogeneous spaces. These results have
been established for spherical homogeneous spaces over p-adic fields, but
most of their analogs exist for automorphic functions, and the talk will
attempt to cover those as well (especially in the function field case,
where technicalities due to Archimedean places do not arise).
The general structure of these results is the following: For a given
G-space X, there are "simpler" G-spaces X_i (of the same dimension but
with more symmetries, i.e. non-trivial groups of G-automorphisms) such
that functions on X decompose into a "discrete modulo automorphisms" part
plus a homomorphic image of the "discrete modulo automorphisms" part of
the spaces X_i. There are smooth and L^2 versions of this story, and for
the former the word "discrete" should be replaced by "cuspidal".
The talk will emphasize general principles (largely based on ideas of
Joseph Bernstein) that give rise to the same kind of decomposition
irrespective of the space, as well as points in the method that have still
not been clarified enough.
SPECIAL SEMINAR
on Tuesday (May 19), 1:30 p.m, room E 206.
Pham H. Tiep (University of Arizona).
Bounding character values of finite groups of Lie type.
Abstract
Let G be a finite group of Lie type. In spite of many foundational results
on complex representation theory of G, several questions on character
values still remain open. One such question, essential for various
applications, including random walks and word maps on finite simple
groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and
for all irreducible complex characters \chi of G. In the case of symmetric
groups, this problem has been solved by Larsen and Shalev. We will discuss
recent progress on this problem for finite groups of Lie type, obtained in
joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.
Monday (May 18), 4:30 p.m, room E 206.
Joseph Bernstein (Tel Aviv University).
Convexity and subconvexity bounds for Automorphic Periods
(joint work with A. Reznikov).
Abstract
In my lecture I will discuss basic problems related to bounds on
periods of automorphic functions. One of my goals is to discuss
the insight into these bounds given by relation of periods to special
values of L-functions (for example this predicts some convexity and
subconvexity bounds on periods coming from L-function theory).
I describe a method based on Representation Theory of real groups
that allows to analyze such bounds.
I will concentrate on two very concrete problems.
Let Y be a compact Riemannian surface of constant
curvature -1. A Maass form is a function f on Y that
is an eigenfunction of the Laplace operator D.
Problem 1 (Fourier expansion). Fix a closed geodesic C
in Y , restrict some Maass form f to the circle C and
consider Fourier coefficients a_k of this function.
How to give bounds on these coefficients when k tends
to infinity ?
Problem 2 (Triple product). Let p be a product of two
Maass forms.
How to give bound on the scalar product <p, f> of the
function p with a Maass form f in terms of the
eigenvalue of f when this eigenvalue tends to infinity ?
Monday (May 18), 4:30 p.m, room E 206.
Joseph Bernstein (Tel Aviv University).
Convexity and subconvexity bounds for Automorphic Periods
(joint work with A. Reznikov).
Abstract
In my lecture I will discuss basic problems related to bounds on
periods of automorphic functions. One of my goals is to discuss
the insight into these bounds given by relation of periods to special
values of L-functions (for example this predicts some convexity and
subconvexity bounds on periods coming from L-function theory).
I describe a method based on Representation Theory of real groups
that allows to analyze such bounds.
I will concentrate on two very concrete problems.
Let Y be a compact Riemannian surface of constant
curvature -1. A Maass form is a function f on Y that
is an eigenfunction of the Laplace operator D.
Problem 1 (Fourier expansion). Fix a closed geodesic C
in Y , restrict some Maass form f to the circle C and
consider Fourier coefficients a_k of this function.
How to give bounds on these coefficients when k tends
to infinity ?
Problem 2 (Triple product). Let p be a product of two
Maass forms.
How to give bound on the scalar product <p, f> of the
function p with a Maass form f in terms of the
eigenvalue of f when this eigenvalue tends to infinity ?
SPECIAL SEMINAR
on Tuesday (May 19), 1:30 p.m, room E 206.
Pham H. Tiep (University of Arizona).
Bounding character values of finite groups of Lie type.
Abstract
Let G be a finite group of Lie type. In spite of many foundational results
on complex representation theory of G, several questions on character
values still remain open. One such question, essential for various
applications, including random walks and word maps on finite simple
groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and
for all irreducible complex characters \chi of G. In the case of symmetric
groups, this problem has been solved by Larsen and Shalev. We will discuss
recent progress on this problem for finite groups of Lie type, obtained in
joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.
Thursday (May 21), 4:30 p.m, room E 206. Joseph Bernstein (Tel Aviv University). Convexity and subconvexity bounds of Automorphic Periods. II.
No more meetings of the seminar in this quarter.
As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. We begin on Thursday Oct 1 with Peter Sarnak’s talk on the very important notion of analytic conductor in the theory of automorphic forms, see http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands for more details. On Thursday Oct 8 Laurent Fargues will begin his mini-course “Geometrization of the local Langlands correspondence”, in which he will explain his new and very exciting conjectures. (As far as I understand, they are in the spirit of the global unramified geometric Langalnds conjecture, but instead of usual curves he considers the “curve” that he and Fontaine had associated to an arbitrary non-Archimedean local field E. Because of that, he ends up with conjectures that would imply the local Langlands conjecture for E.) Presumably, Fargues will give 8 lectures. The title and abstract of his course can be found at http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands
Thursday (Oct 1), 4:30 p.m, room E 206.
Peter Sarnak (IAS). The analytic conductor in the theory of automorphic
forms.
Abstract
The analytic conductor of an automorphic cusp form on GL(n) over a number
field is a measure of its complexity: especially in connection with the
corresponding L-function. We review some of the definitions, properties
and central role played by the conductor. If time permits we discuss some
recent applications to fast computations of epsilon factors and the Mobius
function.
No seminar on Monday (Oct.5).
On Thursday (Oct.8) Laurent Fargues will begin his mini-course on
Geometrization of the local Langlands correspondence
Thursday (October 8), 4:30 p.m, room E 206.
Laurent Fargues (Institut de Mathematiques de Jussieu) will begin his
mini-course on
Geometrization of the local Langlands correspondence
I will explain a recent conjecture giving a geometrization of the local
Langlands correspondence over a non-archimedean local field. The purpose
is to explain the precise statement of the conjecture and evidences for
it. For this I will introduce the objects that appear, in particular the
curve defined and studied in our joint work with Fontaine, the structure
of G-bundles on this curve and the basic properties of the associated
stack of G-bundles.
Presumably, there will be 8 lectures in the mini-course.
No seminar on Monday Oct 12. Laurent Fargues will give his second lecture next Thursday (Oct 15). I will inform you when the notes of his first lecture become available. Program for the rest of October: Laurent Fargues will speak on Oct 15, 22, 26. Bhargav Bhatt will give a talk on Oct 19. Information on the Oberwolfach workshop mentioned by Fargues is at www.mfo.de/occasion/1614/www_view
Laurent Fargues says that to understand his yesterday lecture, one can read one of the following articles available at his homepage http://webusers.imj-prg.fr/~laurent.fargues/Publications.html 1. Factorization of analytic functions in mixed characteristic 2. Vector bundles and p-adic Galois representations 3. Vector bundles on curves and p-adic Hodge theory The first article is the most elementary, and Fargues says that already that article covers the material of his yesterday lecture. (The third article is the most advanced.)
The notes of the first lecture by Fargues are at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf The notes were made by Sean Howe. He says: > As the lectures continue I'll update that file. Already in the first > lecture there are probably some typos I've missed; I'll correct them as I > or others find them.
Laurent Fargues will give his second lecture on Thursday (Oct 15), 4:30 p.m, room E 206.
Monday (Oct 19), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian.
Abstract
The affine Grassmannian is an ind-variety over a field k that parametrizes
k[[t]]-lattices in a fixed k((t))-vector space. I will explain the
construction of a p-adic analog, i.e., an ind-scheme over F_p that
parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on
recent work of X. Zhu. The construction takes place in the world of
algebraic geometry with perfect schemes, and a large portion of the talk
will be devoted to explaining certain surprisingly nice features of this
world. (This talk is based on joint work with Peter Scholze.)
Monday (Oct 19), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian.
Abstract
The affine Grassmannian is an ind-variety over a field k that parametrizes
k[[t]]-lattices in a fixed k((t))-vector space. I will explain the
construction of a p-adic analog, i.e., an ind-scheme over F_p that
parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on
recent work of X. Zhu. The construction takes place in the world of
algebraic geometry with perfect schemes, and a large portion of the talk
will be devoted to explaining certain surprisingly nice features of this
world. (This talk is based on joint work with Peter Scholze.)
Laurent Fargues will give his third lecture on Thursday (Oct 22), 4:30 p.m, room E 206.
The notes at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf now include the second lecture by Fargues. Fargues will give his third lecture on Thursday (usual time and place).
Laurent Fargues will give his next lecture on MONDAY (Oct 26), 4:30 p.m, room E 206.
No seminar on Thursday. The notes from Fargues' Thursday lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Laurent Fargues will give his next lecture on Monday (Nov 2), 4:30 p.m, room E 206.
Laurent Fargues will give his next lecture on
Monday (Nov 2), 4:30 p.m, room E 206.
*******
Many of us know the name of Andrei Zelevinsky, who worked in
representation theory (including the Langlands program) and combinatorics.
He died a few years ago, when he was only 60.
I just learned that Northeastern University has established a prestigious
post-doctoral position named in Zelevinsky's memory. More information (in
particular, instructions how to donate) can be found at
http://avzel.blogspot.com/2015/10/andrei-zelevinsky-research-instructor.html
See also
http://zelevinsky.com/Zelevinsky_Fund_Letter.pdf
Andrei was a very good mathematician and a very good man.
(We first met at a mathematical olympiad when he was 16 and I was 15.
At that time he was very impressed by Vilenkin's book on combinatorics.
Later combinatorics became a major area of his research...)
The notes of Fargues' Monday lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Laurent Fargues will give his next lecture tomorrow, i.e., Monday (Nov 2), 4:30 p.m, room E 206.
Laurent Fargues has to CANCEL his lecture today.
Laurent Fargues will give his next lecture on Thursday (Nov 5), 4:30 p.m, room E 206.
Laurent Fargues will give his next lecture on Monday (Nov 9), 4:30 p.m, room E 206.
Laurent Fargues will give his next lecture on Thursday (Nov 12), 4:30 p.m, room E 206.
Monday (Nov 16) and Thursday (Nov 19), 4:30 p.m, room E 206.
Akshay Venkatesh (Stanford University). Motivic cohomology and the
cohomology of arithmetic groups.
Abstract
The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke
action. The same system of Hecke eigenvalues will usually occur in
multiple cohomological degrees. This suggests the existence of extra
endomorphisms of the cohomology that commute with the Hecke operators but
shift cohomological degree. (In the Shimura case these are provided by
"Lefschetz operators" but the situation in general is much more
interesting.)
I will explain a conjecture that, in fact, the motivic cohomology of the
associated motives act on the cohomology and provide these extra
endomorphisms. (According to the Langlands program, a Hecke eigenclass
occurring in cohomology should give a system of motives, indexed by
representations of the dual group. The "associated" motives we need are
the ones associated to the adjoint representation of the dual group.)
This structure should exist at the level of cohomology with
Q-coefficients, but I don't know how to construct it.
However, one can construct the corresponding action on cohomology with
real or p-adic coefficients, using the corresponding regulator map on the
motivic cohomology, and then try to get evidence that it preserves
Q-structures.
LECTURE ONE: I will explain the overall conjecture and how to construct
the action with real coefficients. This is joint work with Prasanna. In
particular, we are able to give evidence that the action preserves
Q-structures, basically by relating it to the theory of periods of
automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture),
and also by using analytic torsion.
LECTURE TWO: I will explain the story with p-adic coefficients. The
story here is algebraically richer; there are two related way of
constructing extra endomorphisms of the cohomology. One is via a derived
version of the Hecke algebra and one is via a derived version of Mazur's
Galois deformation ring (joint with Soren Galatius). Here there is no
evidence, at present, that this preserves Q-structures.
Thursday (Nov 19), 4:30 p.m, room E 206.
Akshay Venkatesh (Stanford University). Motivic cohomology and the
cohomology of arithmetic groups. II.
Abstract
The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke
action. The same system of Hecke eigenvalues will usually occur in
multiple cohomological degrees. This suggests the existence of extra
endomorphisms of the cohomology that commute with the Hecke operators but
shift cohomological degree. (In the Shimura case these are provided by
"Lefschetz operators" but the situation in general is much more
interesting.)
I will explain a conjecture that, in fact, the motivic cohomology of the
associated motives act on the cohomology and provide these extra
endomorphisms. (According to the Langlands program, a Hecke eigenclass
occurring in cohomology should give a system of motives, indexed by
representations of the dual group. The "associated" motives we need are
the ones associated to the adjoint representation of the dual group.)
This structure should exist at the level of cohomology with
Q-coefficients, but I don't know how to construct it.
However, one can construct the corresponding action on cohomology with
real or p-adic coefficients, using the corresponding regulator map on the
motivic cohomology, and then try to get evidence that it preserves
Q-structures.
LECTURE ONE: I will explain the overall conjecture and how to construct
the action with real coefficients. This is joint work with Prasanna. In
particular, we are able to give evidence that the action preserves
Q-structures, basically by relating it to the theory of periods of
automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture),
and also by using analytic torsion.
LECTURE TWO: I will explain the story with p-adic coefficients. The
story here is algebraically richer; there are two related way of
constructing extra endomorphisms of the cohomology. One is via a derived
version of the Hecke algebra and one is via a derived version of Mazur's
Galois deformation ring (joint with Soren Galatius). Here there is no
evidence, at present, that this preserves Q-structures.
Laurent Fargues will speak on Monday (Nov 23), 4:30 p.m, room E 206. Notes from the 5th and 6th lectures by Fargues are now available at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Notes from the 7th lecture are expected to appear relatively soon (maybe on Sunday).
The notes of Fargues' November 12 lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf He will speak today at the usual time and place.
Laurent Fargues will continue tomorrow (i.e., Tuesday Nov 24), 1:30 p.m, room E 202.
No more meetings of the seminar this quarter. Happy Thanksgiving!
The first meeting of the seminar is on
Thursday (Jan 7), 4:30 p.m, room E 206.
Alexander Beilinson will begin his series of talks on
The singular support and characteristic cycle of etale sheaves.
Abstract
Singular support and characteristic cycle are fundamental notions of the
theory of D-modules; they were rendered into the setting of constructible
sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira
in their book on microlocal theory of sheaves. This series of talks treats
the case of etale sheaves on varieties over a field of finite
characteristic studied recently by
T.Saito, http://lanl.arxiv.org/abs/1510.03018
and myself, http://lanl.arxiv.org/abs/1505.06768.
In dimension one the story amounts to the classical
Grothendieck-Ogg-Shafarevich formula.
In the first talk I will remind, as a warm-up, the classical D-module and
Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar
old observation by Deligne about non-integrability of characteristics, and
introduce the main tool of Brylinski's Radon transform. If time permits, I
will give a proof of the basic upper estimate on the dimension of the
singular support.
Thursday (Jan 7), 4:30 p.m, room E 206.
Alexander Beilinson will begin his series of talks on
The singular support and characteristic cycle of etale sheaves.
Abstract
Singular support and characteristic cycle are fundamental notions of the
theory of D-modules; they were rendered into the setting of constructible
sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira
in their book on microlocal theory of sheaves. This series of talks treats
the case of etale sheaves on varieties over a field of finite
characteristic studied recently by
T.Saito, http://lanl.arxiv.org/abs/1510.03018
and myself, http://lanl.arxiv.org/abs/1505.06768.
In dimension one the story amounts to the classical
Grothendieck-Ogg-Shafarevich formula.
In the first talk I will remind, as a warm-up, the classical D-module and
Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar
old observation by Deligne about non-integrability of characteristics, and
introduce the main tool of Brylinski's Radon transform. If time permits, I
will give a proof of the basic upper estimate on the dimension of the
singular support.
Deligne's letter mentioned by Sasha is at
http://math.uchicago.edu/~drinfeld/Deligne's_letter_SingSupp.pdf
******
Next talk:
Monday (Jan 11), 4:30 p.m, room E 206.
Daniil Rudenko (Moscow). Goncharov conjectures and functional equations
for polylogarithms.
Abstract
Classical polylogarithms and functional equations which these functions
satisfy have been studied since the beginning of the 19th century.
Nevertheless, the structure of these equations is still understood very
poorly. I will explain an approach to this subject, based on the link
between polylogarithms and mixed Tate motives.
A substantial part of the talk will be devoted to the explanation of this
link, provided by Goncharov Conjectures. After that, I will present some
results about functional equations which can be proved unconditionally. If
time permits, I will finish with another application of this circle of
ideas to scissor congruence theory.
A few years ago I wrote some notes for myself, which can be found here:
http://www.math.uchicago.edu/~drinfeld/Cotangent_notes-2011/Notes-2011.pdf
They are closely related to Sasha's Thursday talk. A brief explanation of
the subject of my notes and the relation with Sasha's talk can be found
here:
http://www.math.uchicago.edu/~drinfeld/Cotangent_notes-2011/Read_me.pdf
******
As already announced, on Monday (Jan 11) Daniil Rudenko will speak on
Goncharov conjectures and functional equations for polylogarithms.
Thursday (Jan 7), 4:30 p.m, room E 206. Alexander Beilinson will give his second talk on The singular support and characteristic cycle of etale sheaves.
Thursday (Jan 14), 4:30 p.m, room E 206. Alexander Beilinson will give his second talk on The singular support and characteristic cycle of etale sheaves.
No seminar on Monday. Beilinson will continue on Thursday (Jan 21).
Beilinson will continue on Monday (Jan 25).
Sasha will speak on Monday (as announced before). Here is Sean Howe's message about his notes of Fargues' lectures. -------------------------------------------------------------------------- Subject: Full notes available From: "Sean Howe" <seanpkh@gmail.com> Date: Sat, January 23, 2016 1:12 pm To: "Vladimir G. Drinfeld" <drinfeld@math.uchicago.edu> -------------------------------------------------------------------------- Hi all, The full notes from Fargues' lectures last quarter are now available on my website: http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf I am sorry for the very long delay in posting the final two lectures; I had some unresolved questions about the last lecture and had trouble finding the time to understand them over winter break. The notes are a preliminary version still -- eventually I will add more references, clean up typos, and most importantly fix some remaining issues with the final section on local-global compatibility (which is still very rough -- I've put a warning about this in the notes). But I thought it'd be better to get them up now and then worry about that in the future! Please let me know of any changes that should be made and I will try to incorporate them in a more timely fashion. Thanks! Best, Sean
Beilinson will continue on Thursday (Jan 28).
No seminar on Monday. The next meeting is on Thursday.
The title and abstract of the talk are below.
("Schober" is a German word; one of its meanings is "haystack". My guess
is that the Schobers from Kapranov's talk are "kind of stacks".)
Thursday (Feb 4), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan). Perverse Schobers.
Abstract
I will explain the project of developing a theory of "perverse sheaves of
triangulated categories". One motivation for it is the desire of
introducing coefficients in the definition of Fukaya categories (which are
categorical analogs of (co)homology with constant coefficients). Since
perverse sheaves are complexes of sheaves and not just sheaves, their
categorical analogs are not obvious. Nevertheless in several interesting
cases the definition can be made, and we can make the first steps in the
cohomological formalism. The talk is based on joint work with T.
Dyckerhoff, V. Schechtman and Y. Soibelman.
Thursday (Feb 4), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan). Perverse Schobers.
Abstract
I will explain the project of developing a theory of "perverse sheaves of
triangulated categories". One motivation for it is the desire of
introducing coefficients in the definition of Fukaya categories (which are
categorical analogs of (co)homology with constant coefficients). Since
perverse sheaves are complexes of sheaves and not just sheaves, their
categorical analogs are not obvious. Nevertheless in several interesting
cases the definition can be made, and we can make the first steps in the
cohomological formalism. The talk is based on joint work with
T.Dyckerhoff, V.Schechtman and Y.Soibelman.
No seminar on Monday.
Thursday (Feb 11), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).
Homotopy Lie algebras associated to secondary polytopes.
Abstract
Motivated by the work of Gaiotto-Moore-Witten on the "algebra of
infrared", we construct a homotopy Lie algebra associated to the secondary
polytope of a finite set A of points in the n-dimensional Euclidean
space. While the construction can be made for any n (and leads to
E_n-algebras), the case of "physical" interest is when A consists of
critical values of a holomorphic Morse function. The talk is based on
joint work with M. Kontsevich and Y. Soibelman.
Thursday (Feb 11), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).
Homotopy Lie algebras associated to secondary polytopes.
Abstract
Motivated by the work of Gaiotto-Moore-Witten on the "algebra of
infrared", we construct a homotopy Lie algebra associated to the secondary
polytope of a finite set A of points in the n-dimensional Euclidean
space. While the construction can be made for any n (and leads to
E_n-algebras), the case of "physical" interest is when A consists of
critical values of a holomorphic Morse function. The talk is based on
joint work with M. Kontsevich and Y. Soibelman.
Monday (Feb 15), 4:30 p.m, room E 206.
Kiran Kedlaya (UCSD). Introduction to F-isocrystals. I
Prof. Kedlaya kindly agreed to give two introductory lectures on this
important subject (the second one on March 4).
Abstract
Let X be a variety over a field of characteristic p>0. The notion of
l-adic local system on X has not one but two p-adic analogs, called
"convergent F-isocrystal" and "overconvergent F-isocrystal". I will start
from scratch and give an overview of the theory of F-isocrystals, paying
close attention to analogies from the l-adic case.
Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202. No meetings until then. Kedlaya's notes on F-isocrystals are here: http://kskedlaya.org/papers/isocrystals.pdf
Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202. (Unusual day, time, and room!) Kedlaya's notes on F-isocrystals are updated: http://kskedlaya.org/papers/isocrystals.pdf
Kiran Kedlaya will speak tomorrow (Friday) at 4:00 p.m. in room E202. (Unusual day, time, and room!)
No seminar on Monday.
Thursday (March 10), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric
Langlands correspondence.
Abstract
The classical limit of the geometric Langlands correspondence is the
conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a
reductive group G and its dual. There was a significant progress on this
statement when G=GL(n) and the Hitchin fibers are identified with
compacitified Jacobians of spectral curves. Unfortunately, the methods are
specific to the group GL(n), and much less is known about the case of
general G.
In my talk, I plan to review the current state of the area, and then
sketch a new approach (work in progress joint with R.Fedorov). The
approach is based on studying the classical limit of the Hecke `algebra',
which turns out to be a much richer object than its usual (`quantum')
counterpart.
Thursday (March 10), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric
Langlands correspondence.
Abstract
The classical limit of the geometric Langlands correspondence is the
conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a
reductive group G and its dual. There was a significant progress on this
statement when G=GL(n) and the Hitchin fibers are identified with
compacitified Jacobians of spectral curves. Unfortunately, the methods are
specific to the group GL(n), and much less is known about the case of
general G.
In my talk, I plan to review the current state of the area, and then
sketch a new approach (work in progress joint with R.Fedorov). The
approach is based on studying the classical limit of the Hecke `algebra',
which turns out to be a much richer object than its usual (`quantum')
counterpart.
No more meetings this quarter.
No meetings this week and the next one. On April 11 (Monday) Kazuya Kato will begin his series of talks.
Monday (April 11), 4:30 p.m, room E 206.
Kazuya Kato. On compactifications of
the moduli spaces of Drinfeld modules. I.
Abstract
The main subject in my talk is to construct
(1) toroidal compactifications of the moduli spaces of Drinfeld modules.
This is similar to the well known
(2) toroidal compactifications of the moduli spaces of polarized abelian
vareities.
But there is a big difference as is explained below.
This is a joint work with T. Fukaya and R. Sharifi. We are studying the
analogue of Sharifi conjecture (it is a refined Iwasawa theory) for GL(n)
of a function field, and then the nice toroidal boundary of the moduli
space of Drinfeld modules became necessary.
The moduli space of polarized abelian varieties has two kinds of
compactification, Satake-Baily-Borel compactification and toroidal
compactifications. For the moduli space of Drinfeld modules, an
analogue of the Satake-Baily-Borel compactification was constructed by
Kapranov and Pink. The analogy is very strong here. Pink wrote a short
paper in 1994 on toroidal compactifications of the moduli space of
Drinfeld modules. But the details are not yet published.
For the toroidal compactification, there is a big difference between (1)
and (2). These toroidal compactifications treat degenerations of Drinfeld
modules and of polarized abelian varieties, respectively. In the
degeneration, the local monodromy of a degenerating polarized abelian
variety has length of unipotence two, but the local monodromy
of a degenerating Drinfeld module can have bigger length of unipotence. In
my talk,
1. I give an overview of the analytic theory and explain how such
difference of (1) and (2) appears.
2. I explain the theory of degeneration of Drinfeld modules (theory of
iterated Tate uniformizations, where the iteration is necessary to treat
the larger length of the unipotence).
To my regret, these 1 and 2 are the best things which I can explain well
now. We have not yet completed a paper on the construction of the toroidal
compactifications and we are not yet perfectly sure that the proofs are
OK. I hope to explain the construction in a next opportunity.
No seminar on Thursday. Kazuya Kato will continue on Monday April 18.
Monday (April 18), 4:30 p.m, room E 206.
Kazuya Kato. On compactifications of the
moduli spaces of Drinfeld modules. II.
Abstract
An elliptic curve over \C is presented as \C/L where L is a \Z-lattice of
rank 2. Tate discovered that an elliptic curve over a local field K has
the presentation K^\times/q^{\Z} as the quotient of the multiplicative
group of K by a \Z-lattice q^{\Z} of rank 1 if the elliptic curve has
multiplicative reduction. This was generalized by Raynaud, Mumford,
Faltings-Chai to higher dimensional abelian varieties over the quotient
field of a complete local normal integral domain of higher dimension.
In his first paper on Drinfeld modules, Drinfeld proved that the Drinfeld
module over a local filed K has a similar presentation K/L as the
quotient of the additive group K by a certain lattice L.
I will explain how to generalize this theory of Drinfeld to the higher
dimensional base case.
This is a theory of degeneration of Drinfeld modules. This is important
for the toroidal compactification of the moduli space of Drinfeld
modules. The q of Tate in the case of an elliptic curve is the best
coordinate function of the compactified modular curve at the cusp, but q
is not an algebraic function. It is an analytic function or a function
which appears after the completion. The best coordinate functions of the
toroidal compactification at the boundary are not algebraic, and they
appear to classify the lattice L which appears after the completion.
There is a big difference from the case of abelian variety. This
difference is due to the fact that the length of the unipotence of the
local monodromy is two for abelian varieties but can be bigger for
Drinfeld modules.
No seminar on Thursday April 21.
Thursday (April 28), 4:30 p.m, room E 206.
Tsao-Hsien Chen (NWU) Quantization of Hitchin integrable system via
positive characteristic.
Abstract
In their work
"Quantization of Hitchin's integrable system and Hecke eigensheaves",
Beilinson and Drinfeld give a construction of an automorphic D-module
corresponding to a local system which carries an additional structure of
an oper.
In my talk, I will explain a new proof of this result, in the case of
G=GL(n), via positive characteristic method. This talk is based on joint
work with R.Bezrukavnikov, R.Travkin and X.Zhu.
No seminar on Monday.
****
Thursday (May 5), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. I.
Abstract
In his recent article "The characteristic cycle and the singular support
of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the
theory of characteristic cycle that generalizes the theory of
Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties
over a field of arbitrary characteristic. For sheaves on a curve the
characteristic cycle amounts to the Artin conductor. One of the central
results of the theory is the global Euler characteristic formula; for a
curve this is the classical formula of Grothendieck-Ogg-Shafarevich.
In the talks I will explain the principal ideas of Saito's theory and
sketch the proofs. They continue my winter talks about the singular
support; all necessary facts will be reminded.
Thursday (May 5), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. I.
Abstract
In his recent article "The characteristic cycle and the singular support
of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the
theory of characteristic cycle that generalizes the theory of
Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties
over a field of arbitrary characteristic. For sheaves on a curve the
characteristic cycle amounts to the Artin conductor. One of the central
results of the theory is the global Euler characteristic formula; for a
curve this is the classical formula of Grothendieck-Ogg-Shafarevich.
In the talks I will explain the principal ideas of Saito's theory and
sketch the proofs. They continue my winter talks about the singular
support; all necessary facts will be reminded.
Presumably, Beilinson will continue on May 16.
Next week there will be two talks by Yun (Monday and Thursday).
*****
Monday (May 9), 4:30 p.m, room E 206.
Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of
L-functions for function fields. I.
Abstract
In joint work with Wei Zhang, we prove a higher derivative analogue of the
Waldspurger formula and the Gross-Zagier formula in the function field
setting under some unramifiedness assumptions. Our formula relates the
self-intersection number of certain cycles on the moduli of Drinfeld
Shtukas for GL(2) to higher derivatives of
automorphic L-functions for GL(2).
In the first talk I will give motivation and state the main results,
giving all the necessary definitions. In the second talk (Thursday May 12)
I will sketch the geometric ideas in the proof.
Thursday (May 12), 4:30 p.m, room E 206. Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of L-functions for function fields. II.
Monday (May 16), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. II.
Abstract
I will sketch the proofs of the theorems on characteristic cycles of etale
sheaves on varieties over a field of arbitrary characteristic.
Thursday (May 19), 4:30 p.m, room E 206. Alexander Beilinson. The characteristic cycle of an etale sheaf. III.
Monday (May 23), 4:30 p.m, room E 206.
Sam Raskin (MIT). Single variable calculus and local geometric Langlands
Abstract
The moduli of (possibly irregular) formal connections in one variable (up
to gauge transformations) is an infinite-dimensional space that "feels
finite-dimensional," e.g., it has finite-dimensional tangent spaces.
However, it is not so clear how this perception is actually reflected in
the global geometry of this space.
Previous works have focused on explicit parametrization of this space. As
we will recall during the talk, this approach has significant limitations,
and is insufficient to say anything about the global geometry. But we will
instead find that this space appears kinder through the lens of
homological (or more poetically, noncommutative) geometry, exhibiting
better features than all its close relatives.
Finally, we will discuss how these results lend credence to the existence
of a de Rham Langlands program incorporating arbitrary singularities (the
usual story is unramified, or at worst has Iwahori ramification).
Thursday (May 26), 4:30 p.m, room E 206.
Sam Raskin (MIT). W-algebras and Whittaker categories
Abstract
Affine W-algebras are a somewhat complicated family of (topological)
associative algebras associated with a semisimple Lie algebra, quantizing
functions on the algebraic loop space of Kostant's slice. They have
attracted a great deal of attention because of Feigin-Frenkel's duality
theorem for them, which identifies W-algebras for a Lie algebra and for
its Langlands dual through a subtle construction.
The purpose of this talk is threefold: 1) to introduce a natural family of
(discrete) modules for the affine W-algebra, 2) to prove an analogue of
Skryabin's equivalence in this setting, realizing the category of
(discrete) modules over the W-algebra in a more natural way, and 3) to
explain how these constructions help understand Whittaker categories in
the more general setting of local geometric Langlands. These three points
all rest on a simple geometric observation, which provides a family of
affine analogues of Bezrukavnikov-Braverman-Mirkovic.
No more meetings of the seminar this quarter.
No meetings this week.
The first meeting is on Oct. 3 (Monday), 4:30 p.m, room E 206.
I will discuss some recent results by K.Kedlaya and me, see
http://arxiv.org/abs/1604.00660
Let X be a variety over F_p . Fix a prime \ell different from p, an
algebraic closure \overline{Q_\ell}, and a p-adic valuation v of the
subfield of algebraic numbers in \overline{Q_\ell} normalized so that
v(p)=1. Let M be a \overline{Q_\ell}-sheaf on X such that for every closed
point x in X the eigenvalues of the geometric Frobenius acting on M_x are
algebraic numbers. Applying the valuation v to these numbers and dividing
by the degree of v over F_p, we get a collection of rational numbers,
which are called slopes of M at x.
We proved that if X is a smooth curve and M is an irreducible local system
then for almost all x the gaps between two consecutive slopes of M at x
are not greater than 1. If M has rank 2 then one can even replace “almost
all” by “all”, but in the rank 3 case this is false.
I will say almost nothing about the proof (which is based on the theory of
F-isocrystals).
Monday (Oct 3), 4:30 p.m, room E 206.
V.Drinfeld. Slopes of irreducible local systems.
Abstract
I will discuss some recent results by K.Kedlaya and me, see
http://arxiv.org/abs/1604.00660
Let X be a variety over F_p . Fix a prime \ell different from p, an
algebraic closure \overline{Q_\ell}, and a p-adic valuation v of the
subfield of algebraic numbers in \overline{Q_\ell} normalized so that
v(p)=1. Let M be a \overline{Q_\ell}-sheaf on X such that for every closed
point x in X the eigenvalues of the geometric Frobenius acting on M_x are
algebraic numbers. Applying the valuation v to these numbers and dividing
by the degree of x over F_p, we get a collection of rational numbers,
which are called slopes of M at x.
We proved that if X is a smooth curve and M is an irreducible local system
then for almost all x the gaps between two consecutive slopes of M at x
are not greater than 1. If M has rank 2 then one can even replace “almost
all” by “all”, but in the rank 3 case this is false.
I will say almost nothing about the proof (which is based on the theory of
F-isocrystals).
No seminar on Thursday (Oct 6).
Monday (Oct 10), 4:30 p.m, room E 206.
Ben Davison (IST, Vienna). Purity and surjectivity for some symplectic
stacks.
Abstract
I will explain a special case of a general procedure, called dimensional
reduction, for identifying the cohomology of the stack-theoretic zero
locus of a moment map with the vanishing cycle cohomology of the stack of
objects in a 3-Calabi-Yau category, focussing almost entirely on down to
earth cases coming from the theory of representations of quivers. I will
also introduce another useful feature of this theory, which is the
observation that the map from the stack of representations to the coarse
moduli space can be "approximated by proper maps" - this amounts to a nice
(module-theoretic) partial compactification of Totaro's construction for
calculating equivariant cohomology. I will explain how these two features
together give rise to surprising purity and surjectivity results.
Monday (Oct 10), 4:30 p.m, room E 206.
Ben Davison (IST, Vienna). Purity and surjectivity for some symplectic
stacks.
Abstract
I will explain a special case of a general procedure, called dimensional
reduction, for identifying the cohomology of the stack-theoretic zero
locus of a moment map with the vanishing cycle cohomology of the stack of
objects in a 3-Calabi-Yau category, focussing almost entirely on down to
earth cases coming from the theory of representations of quivers. I will
also introduce another useful feature of this theory, which is the
observation that the map from the stack of representations to the coarse
moduli space can be "approximated by proper maps" - this amounts to a nice
(module-theoretic) partial compactification of Totaro's construction for
calculating equivariant cohomology. I will explain how these two features
together give rise to surprising purity and surjectivity results.
Thursday (Oct 13), 4:30 p.m, room E 206.
David Nadler (Berkeley). Arboreal singularities of Lagrangian
subvarieties. I.
Abstract
Arboreal singularities are a class of singularities of Lagrangian
subvarieties of symplectic manifolds. They have several elementary
characterizations and are essentially combinatorial objects. It turns out
that any Lagrangian singularity admits a deformation to a nearby
Lagrangian subvariety with arboreal singularities. Moreover, the
deformation is non-characteristic in the sense that categories of
"Lagrangian branes", for example in the form of microlocal sheaves, are
invariant under it. This yields an elementary method for calculating them
which can be applied in situations of interest in mirror symmetry. At a
more basic level, Lagrangian subvarieties with arboreal singularities
offer a higher dimensional analogue of ribbon graphs from which one can
hope to recover the ambient symplectic manifold itself.
The talks will be based primarily on the papers: arXiv:1309.4122,
arXiv:1507.01513, arXiv:1507.08735, and time permitting, work in progress
with Eliashberg and Starkston.
Monday (Oct 17), 4:30 p.m, room E 206.
David Nadler (Berkeley). Arboreal singularities of Lagrangian
subvarieties. II.
Abstract
Arboreal singularities are a class of singularities of Lagrangian
subvarieties of symplectic manifolds. They have several elementary
characterizations and are essentially combinatorial objects. It turns out
that any Lagrangian singularity admits a deformation to a nearby
Lagrangian subvariety with arboreal singularities. Moreover, the
deformation is non-characteristic in the sense that categories of
"Lagrangian branes", for example in the form of microlocal sheaves, are
invariant under it. This yields an elementary method for calculating them
which can be applied in situations of interest in mirror symmetry. At a
more basic level, Lagrangian subvarieties with arboreal singularities
offer a higher dimensional analogue of ribbon graphs from which one can
hope to recover the ambient symplectic manifold itself.
The talks will be based primarily on the papers: arXiv:1309.4122,
arXiv:1507.01513, arXiv:1507.08735, and time permitting, work in progress
with Eliashberg and Starkston.
NB: the next meeting is on *Friday* (not Thursday)!
Friday (Oct 21), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). Metaplectic geometric Langlands.
Abstract
We will explain what it is the geometric counterpart of a metaplectic
extension (in the global or local cases), the construction of the
metaplectic spectral group and metaplectic geometric Satake.
The goal of the talk is to state the metaplectic spectral decomposition
conjecture, which is a metaplectic analog of the statement that D(Bun_G)
is acted on by the monoidal category QCoh(LocSys_{G^L}).
This is a joint work with S. Lysenko.
NB: the next meeting is on *Friday* (not Thursday)!
Friday (Oct 21), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). Metaplectic geometric Langlands.
Abstract
We will explain what it is the geometric counterpart of a metaplectic
extension (in the global or local cases), the construction of the
metaplectic spectral group and metaplectic geometric Satake.
The goal of the talk is to state the metaplectic spectral decomposition
conjecture, which is a metaplectic analog of the statement that D(Bun_G)
is acted on by the monoidal category QCoh(LocSys_{G^L}).
This is a joint work with S. Lysenko.
Monday (Oct 24), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). Quantum geometric Langlands.
Abstract
We will state the quantum global unramified geometric Langlands conjecture
about the equivalence of the twisted D(Bun_G) and D(Bun_{G^L}) for a
matching pair of twistings. We will relate this to the metaplectic
spectral decomposition conjecture from the previous talk. We will explain
how as an ingredient of the quantum global conjecture we have the quantum
local equivalence, between the twisted Whittaker category on the affine
Grassmannian for G, and the Kazhdan-Lusztig category for G^L.
No seminar on Thursday Oct 27.
On Monday Oct 31 and Thursday Nov 3 there will be talks by
Xinwen Zhu (Caltech).
Title of his talks: Towards a p-adic non-abelian Hodge theory.
Abstract
I will first review the non-abelian Hodge theory for complex manifolds and
describe a conjectural p-adic analogue. Then I will discuss what we know
so far and the following consequence of (the known part of) the theory:
Let L be a p-adic local system on a (geometrically) connected algebraic
variety over a number field. If its stalk at one point, regarded as a
p-adic Galois representation, is geometric in the sense of Fontaine-Mazur,
then the stalk at every point is geometric.
This is based on a joint work with Ruochuan Liu.
Monday (Oct 31), 4:30 p.m, room E 206.
Xinwen Zhu (Caltech). Towards a p-adic non-abelian Hodge theory.I.
Abstract
I will first review the non-abelian Hodge theory for complex manifolds and
describe a conjectural p-adic analogue. Then I will discuss what we know
so far and the following consequence of (the known part of) the theory:
Let L be a p-adic local system on a (geometrically) connected algebraic
variety over a number field. If its stalk at one point, regarded as a
p-adic Galois representation, is geometric in the sense of Fontaine-Mazur,
then the stalk at every point is geometric.
This is based on a joint work with Ruochuan Liu.
Thursday (Nov 3), 4:30 p.m, room E 206.
Xinwen Zhu (Caltech). Towards a p-adic non-abelian Hodge theory.II.
Abstract
I will first review the non-abelian Hodge theory for complex manifolds and
describe a conjectural p-adic analogue. Then I will discuss what we know
so far and the following consequence of (the known part of) the theory:
Let L be a p-adic local system on a (geometrically) connected algebraic
variety over a number field. If its stalk at one point, regarded as a
p-adic Galois representation, is geometric in the sense of Fontaine-Mazur,
then the stalk at every point is geometric.
This is based on a joint work with Ruochuan Liu.
Monday (Nov 7), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan). The direct summand conjecture
and its derived variant.
Abstract
In the late 60's, Hochster formulated the direct summand conjecture (DSC)
in commutative algebra, which is the following innocuous looking
assertion: a finite extension A --> B of commutative rings admits an
A-module splitting if A is regular and noetherian. A few years later,
Hochster himself proved the DSC when the ring contains a field; this and
related ideas eventually had a significant impact on the development of
the theory of F-singularities.
In the mixed characteristic setting, the case of dimension <= 3 was
settled by Heitmann in the 90's. The general case, however, remained wide
open until extremely recently, when it was resolved beautifully by Andr\'e
using the theory of perfectoid spaces.
In this talk, I'll discuss a proof of the DSC that is related to, but
simplifies, Andr\'e’s proof. I will also explain why similar ideas help
establish a derived variant of the DSC put forth by de Jong; the latter
roughly states that regular rings have rational singularities. One of my
main goals in this talk to explain why passing from a mixed characteristic
ring to a perfectoid extension is a useable analog of the passage to the
perfection (direct limit over Frobenius) in characteristic p.
The relevant background from perfectoid geometry will be explained.
No seminar on Thursday.
******
Monday (Nov 14), 4:30 p.m, room E 206.
Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I.
Abstract
Given a linear category over a finite field such that the moduli space of
its objects is a smooth Artin stack (and some additional conditions) we
give a formula for the number of absolutely indecomposable objects of the
category and also a similar expression for a certain stacky exponential
sum over the set of all objects of the category. Our formulas involve the
geometry of the cotangent bundle on the moduli stack. The first formula
was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas.
It provides a new approach for counting absolutely indecomposable quiver
representations, vector bundles with parabolic structure on a projective
curve, and irreducible l-adic local systems (via a result of Deligne). Our
second formula resembles formulas appearing in the theory of
Donaldson-Thomas invariants.
Monday (Nov 14), 4:30 p.m, room E 206.
Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I.
Abstract
Given a linear category over a finite field such that the moduli space of
its objects is a smooth Artin stack (and some additional conditions) we
give a formula for the number of absolutely indecomposable objects of the
category and also a similar expression for a certain stacky exponential
sum over the set of all objects of the category. Our formulas involve the
geometry of the cotangent bundle on the moduli stack. The first formula
was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas.
It provides a new approach for counting absolutely indecomposable quiver
representations, vector bundles with parabolic structure on a projective
curve, and irreducible l-adic local systems (via a result of Deligne). Our
second formula resembles formulas appearing in the theory of
Donaldson-Thomas invariants.
Monday (Nov 14), 4:30 p.m, room E 206.
Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I.
Abstract
Given a linear category over a finite field such that the moduli space of
its objects is a smooth Artin stack (and some additional conditions) we
give a formula for the number of absolutely indecomposable objects of the
category and also a similar expression for a certain stacky exponential
sum over the set of all objects of the category. Our formulas involve the
geometry of the cotangent bundle on the moduli stack. The first formula
was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas.
It provides a new approach for counting absolutely indecomposable quiver
representations, vector bundles with parabolic structure on a projective
curve, and irreducible l-adic local systems (via a result of Deligne). Our
second formula resembles formulas appearing in the theory of
Donaldson-Thomas invariants.
The article by Dobrovolska, Ginzburg and Travkin is at http://math.uchicago.edu/~drinfeld/counting.pdf No seminar on Thursday. Ginzburg will give his second talk on Monday (Nov 21), 4:30 p.m, room E 206.
Monday (Nov 21), 4:30 p.m, room E 206.
Victor Ginzburg. Point counting on moduli spaces via factorization
sheaves.II.
Abstract
Given a linear category over a finite field such that the moduli space of
its objects is a smooth Artin stack (and some additional conditions) we
give a formula for the number of absolutely indecomposable objects of the
category and also a similar expression for a certain stacky exponential
sum over the set of all objects of the category. Our formulas involve the
geometry of the cotangent bundle on the moduli stack. The first formula
was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas.
It provides a new approach for counting absolutely indecomposable quiver
representations, vector bundles with parabolic structure on a projective
curve, and irreducible l-adic local systems (via a result of Deligne). Our
second formula resembles formulas appearing in the theory of
Donaldson-Thomas invariants.
No more meetings this quarter. Happy Thanksgiving!
The first meeting is on Jan 9 (Monday), 4:30 p.m, room E 206.
Yiannis Sakellaridis (Rutgers) will begin his series of lectures
“Construction of automorphic L-functions”.
The abstract of the whole course is below.
As far as I understand, the main keyword of this course is “spherical
variety”.
As far as I understand, the course is mostly on the classical (rather than
geometric) Langlands program, but in the case of function fields there
should definitely be a geometric version (with sheaves rather than
functions).
Abstract
L-functions are among the most mysterious objects in number theory. In
particular, they are defined in a very abstract way, but any time we can
say something useful about them we use a geometric or harmonic-analytic
way to represent them. In this series of lectures I will survey different
methods for producing L-functions out of automorphic forms, such as by
period and Rankin-Selberg integrals, including conjectures and explicit
calculations which have some geometric significance that has not been
completely understood.
Monday (Jan 9), 4:30 p.m, room E 206.
Yiannis Sakellaridis (Rutgers). Construction of automorphic L-functions.I.
This is the first lecture of a course. Here is the abstract of the whole
course:
Abstract
L-functions are among the most mysterious objects in number theory. In
particular, they are defined in a very abstract way, but any time we can
say something useful about them we use a geometric or harmonic-analytic
way to represent them. In this series of lectures I will survey different
methods for producing L-functions out of automorphic forms, such as by
period and Rankin-Selberg integrals, including conjectures and explicit
calculations which have some geometric significance that has not been
completely understood.
The next meeting is on Monday Jan 16. (Sakellaridis will give his second lecture.)
Sakellaridis will give his second lecture on Monday (Jan 16), 4:30 p.m, room E 206. Note that our buildings WILL BE LOCKED this Monday because of the holiday. So if you have a UofC card please be sure to have it with you. If you don't then please arrive 10 minutes before the seminar, and somebody will let you in.
The next meeting is on Monday Jan 23. (Sakellaridis will give his next lecture.)
Sakellaridis will give his next lecture on Monday (Jan 23), 4:30 p.m, room E 206.
The next meeting is on Monday Jan 30. (Sakellaridis will give his next lecture.)
Sakellaridis will give his next lecture on Monday (Jan 30), 4:30 p.m, room E 206.
The next meeting is on Monday Feb 6. (Sakellaridis will give his next lecture.)
Sakellaridis will give his next lecture on Monday (Feb 6), 4:30 p.m, room E 206.
The next meeting is on Monday Feb 13. (Sakellaridis will give his next lecture.)
Sakellaridis will give his lecture today at 4:30 p.m, room E 206.
Sakellaridis will give his next lecture on Monday (Feb 20), 4:30 p.m, room E 206.
Sakellaridis will give his lecture tomorrow (Monday) at 4:30 p.m, room E 206.
Sakellaridis will give his next lecture on Monday (Feb 27), 4:30 p.m, room E 206.
Sakellaridis will give his next lecture on Monday (Feb 27), 4:30 p.m, room E 206.
Sakellaridis will give his next lecture on Monday (March 6), 4:30 p.m, room E 206.
Sakellaridis will give his next lecture on tomorrow (Monday March 6), 4:30 p.m, room E 206.
No more meetings of the seminar this quarter.
The first meeting is on April 3 (Monday), 4:30 p.m, room E 206.
Beilinson will give a talk; the title and abstract are below.
********
As already announced, Yiannis Sakellaridis will be giving a course on the
Relative Trace Formula every Wednesday 2:30-4:30 (room E308) starting from
this Wednesday, March 29.
********
Title of Beilinson’s talk on April 3:
Wild ramification and the Euler characteristic.
Abstract
Let F be a constructible sheaf on a variety X over an algebraically closed
field k. If char k = 0 then the global Euler characteristic \chi (X,F)
(as well as its local avatar, the characteristic cycle CC(F)) is
determined by the constructible function x \mapsto rk (F_x ), x\in X(k).
This is no longer true if char k > 0: indeed, if X is a curve then the
Grothendieck-Ogg-Shafarevich formula for \chi (X,F) includes extra local
terms - the conductors that measure the wild ramification.
I will talk about the works
"Wild ramification determines the characteristic cycle" by T.Saito and
Y.Yatagawa (arXiv:1604.01513) and
"Wild ramification and restriction to curves" by H.Kato
(arXiv:1611.07642),
which treat the case dim X > 1. In particular they show that \chi (X,F)
(and CC(F)) are determined by the conductors of the pullback of F to every
curve.
No prior knowledge of the subject (in particular, what is CC(F)) is assumed.
Today (Monday), 4:30 p.m, room E 206.
A. Beilinson. Wild ramification and the Euler characteristic.
Abstract
Let F be a constructible sheaf on a variety X over an algebraically closed
field k. If char k = 0 then the global Euler characteristic \chi (X,F)
(as well as its local avatar, the characteristic cycle CC(F)) is
determined by the constructible function x \mapsto rk (F_x ), x\in X(k).
This is no longer true if char k > 0: indeed, if X is a curve then the
Grothendieck-Ogg-Shafarevich formula for \chi (X,F) includes extra local
terms - the conductors that measure the wild ramification.
I will talk about the works
"Wild ramification determines the characteristic cycle" by T.Saito and
Y.Yatagawa (arXiv:1604.01513) and
"Wild ramification and restriction to curves" by H.Kato
(arXiv:1611.07642),
which treat the case dim X > 1. In particular they show that \chi (X,F)
(and CC(F)) are determined by the conductors of the pullback of F to every
curve.
No prior knowledge of the subject (in particular, what is CC(F)) is assumed.
No seminar on Thursday.
April 10 (Monday), 4:30 p.m, room E 206. Ngo Bao Chau. Perverse sheaves on formal arc spaces I will try to explain the foundational work of Bouthier and Kazhdan on the concept of perverse sheaves on formal arc spaces. We expect this theory to have applications in harmonic analysis over non-Archimedean local fields. I will discuss some examples.
As already announced, today at 4:30 p.m. (room E206) Ngo Bao Chau will speak on "Perverse sheaves on formal arc spaces" (after a work by Bouthier-Kazhdan). ******* Tomorrow Nicolas Templier will give a talk at the NT seminar (see below). His talk is related to the geometric Langlands program. Presumably, he will not finish his talk on Tuesday; in this case he will continue on Thursday at the geometric Langlands seminar (4:30 p.m., room E206). Tuesday April 11, 1:30pm - 3:00pm in Eckhart 206 Nicolas Templier (Cornell): Mirror symmetry for minuscule flag varieties. We prove cases of Rietsch mirror conjecture that the Dubrovin-Givental quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. The isomorphism then comes from global rigidity results where a Hecke eigenform is determined by its local ramification. We reveal relations with the works of Gross, Frenkel-Gross, Heinloth-Ngo-Yun and Zhu on Kloosterman sheaves. The talk will keep the algebraic geometry prerequisite knowledge to a minimum by introducing the above concepts of "mirror" and "crystal" with the examples of CP^1, projective spaces and quadrics. Work with Thomas Lam.
Thursday (April 13), 4:30 p.m, room E 206. Nicolas Templier (Cornell) will finish the talk on Mirror symmetry for minuscule flag varieties that he started on Tuesday at the NT seminar. ********* On Monday (April 17) Ngo Bao Chau will continue his talk on formal arc spaces.
Monday (April 17), 4:30 p.m, room E 206. Ngo Bao Chau will finish his talk on formal arc spaces. The notes of his talks are available at https://math.uchicago.edu/~ngo/Weierstrass.pdf He may keep updating them.
Thursday (April 20), 4:30 p.m, room E 206.
V.Drinfeld. A stacky approach to formal arcs. I.
The abstract is below. On the other hand, the following 2-page file can
serve as a bridge between Ngo's talks and mine:
http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf
(Reading that file is not necessary, but it can help.)
Abstract
As a complement to Ngo’s talks (and without relying on them), I will
explain the geometric ideas behind the computations in my old work
https://arxiv.org/pdf/math/0203263.pdf
on formal arcs. This will probably take 2 talks.
Main points:
(i) a finite type model for the formal neighborhood of a formal arc in a
variety X is provided by the space of maps from the affine line to X/G,
where G is a smooth groupoid acting on X;
(ii) usually G does not come from a group action (but there are important
cases when it does);
(iii) for a given X there is plenty of smooth groupoids acting on X; one
can construct them using “affine blow-ups”;
(iv) there is a useful notion of the Lie algebroid of a smooth groupoid.
Monday (April 24), 4:30 p.m, room E 206. I will give my second (and hopefully last) talk on the stacky approach to formal arcs. In this talk I will try to explain how to work with smooth groupoids. Here are some related materials (no need to read them now; my Monday talk will be a kind of introduction to these texts): Smooth groupoids on smooth manifolds are discussed in the article https://arxiv.org/abs/math/0611259.pdf by Crainic-Fernandes. In particular, they discuss the notion of Lie algebraic of a Lie groupoid. The “Newton groupoid” (the one which is secretly used in my proof of the Grinberg-Kazhdan theorem) is defined in the following notes: http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Newton%20groupoid.pdf There could be mistakes there. The Newton groupoid is the one that I suggested to guess at the end of my previous write-up, see http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf The video of my yesterday's talk is at https://youtu.be/0bG7V4oavxY My notes of that talk are at http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Talk1.pdf
The notes of my yesterday talk are available at http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Talk2.pdf The Newton groupoid I defined yesterday is the one that was denoted by \Gamma_2 at the end of the file http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf (I didn't give the definition of \Gamma_2 there). ******** No seminar on Thursday April 27 and Monday May 1. After that, there will be talks by Geordie Williamson on May 4,8,11. Title: "Studying the decomposition theorem with integral coefficients". The abstract can be at the seminar calendar http://math.uchicago.edu/research/calendar/ (see the description of Williamson's talk on May 4).
Thursday (May 4), 4:30 p.m, room E 206.
Geordie Williamson (Sydney). Studying the decomposition theorem with
integral coefficients. I.
Abstract
I will explain an approach (via intersection form) to understanding if the
decomposition theorem holds with coefficients in characteristic p fields
and the integers. I am interested in these questions primarily with
applications to modular representation in mind, but they are also
fascinating questions by themselves. My lectures will
cover the following topics:
1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition
theorem (for the direct image of the constant sheaf);
2) torsion in integral intersection cohomology and obstacles to attempting
to carry out the above proof over the integers;
3) what we know and don't know about torsion in the intersection
cohomology of Schubert varieties;
4) applications to modular representation theory of algebraic groups
(Lusztig conjecture, Finkelberg-Mirkovic conjecture).
5) optional final topic: parity sheaves, p-canonical basis, tilting
modules, new character formulas (joint work with Riche and
Achar-Makisumi-Riche).
For students wishing to prepare for the lectures, the following links
might be useful:
Part 1) will roughly following my Seminaire Bourbaki lecture:
http://front.math.ucdavis.edu/1603.09235
https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2
Background for parts 2) - 3) can be found in the following paper (see also
a lecture in Stonybrook):
http://front.math.ucdavis.edu/1512.08295
http://scgp.stonybrook.edu/video_portal/video.php?id=2423
Part 4) will roughly follow my Takagi lectures (especially the second part
on constructible sheaves):
https://arxiv.org/abs/1610.06261
https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be
https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be
Part 5) (if it is given) will follow the following two papers:
https://arxiv.org/abs/0906.2994
http://front.math.ucdavis.edu/1512.08296
Monday (May 8), 4:30 p.m, room E 206.
Geordie Williamson (Sydney). Studying the decomposition theorem with
integral coefficients. II.
Abstract
I will explain an approach (via intersection form) to understanding if the
decomposition theorem holds with coefficients in characteristic p fields
and the integers. I am interested in these questions primarily with
applications to modular representation in mind, but they are also
fascinating questions by themselves. My lectures will
cover the following topics:
1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition
theorem (for the direct image of the constant sheaf);
2) torsion in integral intersection cohomology and obstacles to attempting
to carry out the above proof over the integers;
3) what we know and don't know about torsion in the intersection
cohomology of Schubert varieties;
4) applications to modular representation theory of algebraic groups
(Lusztig conjecture, Finkelberg-Mirkovic conjecture).
5) optional final topic: parity sheaves, p-canonical basis, tilting
modules, new character formulas (joint work with Riche and
Achar-Makisumi-Riche).
For students wishing to prepare for the lectures, the following links
might be useful:
Part 1) will roughly following my Seminaire Bourbaki lecture:
http://front.math.ucdavis.edu/1603.09235
https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2
Background for parts 2) - 3) can be found in the following paper (see also
a lecture in Stonybrook):
http://front.math.ucdavis.edu/1512.08295
http://scgp.stonybrook.edu/video_portal/video.php?id=2423
Part 4) will roughly follow my Takagi lectures (especially the second part
on constructible sheaves):
https://arxiv.org/abs/1610.06261
https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be
https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be
Part 5) (if it is given) will follow the following two papers:
https://arxiv.org/abs/0906.2994
http://front.math.ucdavis.edu/1512.08296
Thursday (May 11), 4:30 p.m, room E 206.
Geordie Williamson (Sydney). Studying the decomposition theorem with
integral coefficients. III.
Abstract
I will explain an approach (via intersection form) to understanding if the
decomposition theorem holds with coefficients in characteristic p fields
and the integers. I am interested in these questions primarily with
applications to modular representation in mind, but they are also
fascinating questions by themselves. My lectures will
cover the following topics:
1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition
theorem (for the direct image of the constant sheaf);
2) torsion in integral intersection cohomology and obstacles to attempting
to carry out the above proof over the integers;
3) what we know and don't know about torsion in the intersection
cohomology of Schubert varieties;
4) applications to modular representation theory of algebraic groups
(Lusztig conjecture, Finkelberg-Mirkovic conjecture).
5) optional final topic: parity sheaves, p-canonical basis, tilting
modules, new character formulas (joint work with Riche and
Achar-Makisumi-Riche).
For students wishing to prepare for the lectures, the following links
might be useful:
Part 1) will roughly following my Seminaire Bourbaki lecture:
http://front.math.ucdavis.edu/1603.09235
https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2
Background for parts 2) - 3) can be found in the following paper (see also
a lecture in Stonybrook):
http://front.math.ucdavis.edu/1512.08295
http://scgp.stonybrook.edu/video_portal/video.php?id=2423
Part 4) will roughly follow my Takagi lectures (especially the second part
on constructible sheaves):
https://arxiv.org/abs/1610.06261
https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be
https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be
Part 5) (if it is given) will follow the following two papers:
https://arxiv.org/abs/0906.2994
http://front.math.ucdavis.edu/1512.08296
Monday (May 15), 4:30 p.m, room E 206.
Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. I.
Abstract
The de Rham-Witt complex was constructed by Spencer Bloch in the mid
1970's as a tool to analyze the crystalline cohomology of proper smooth
schemes over a perfect field of characteristic p >0, with its action of
Frobenius, and describe its relations with other types of cohomology, like
Hodge cohomology or Serre's Witt vector cohomology. Since then many
developments have occurred. These lectures are meant as an introduction to
the theory and contain no new material.
After briefly recalling the history of the subject, I will explain the
main construction and in the case of a polynomial algebra give its simple
description by the so-called complex of integral forms. I will then
describe the local structure of the de Rham-Witt complex for smooth
schemes over a perfect field and its application to the calculation of
crystalline cohomology. In the proper smooth case, I will discuss the
slope spectral sequence and the main finiteness properties of the
cohomology of the de Rham-Witt complex in terms of coherent complexes over
the Raynaud ring. I will briefly mention a few complements (logarithmic
Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink
relative variants), and make a tentative list of open problems.
Thursday (May 18), 4:30 p.m, room E 206.
Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. II.
Abstract
The de Rham-Witt complex was constructed by Spencer Bloch in the mid
1970's as a tool to analyze the crystalline cohomology of proper smooth
schemes over a perfect field of characteristic p >0, with its action of
Frobenius, and describe its relations with other types of cohomology, like
Hodge cohomology or Serre's Witt vector cohomology. Since then many
developments have occurred. These lectures are meant as an introduction to
the theory and contain no new material.
After briefly recalling the history of the subject, I will explain the
main construction and in the case of a polynomial algebra give its simple
description by the so-called complex of integral forms. I will then
describe the local structure of the de Rham-Witt complex for smooth
schemes over a perfect field and its application to the calculation of
crystalline cohomology. In the proper smooth case, I will discuss the
slope spectral sequence and the main finiteness properties of the
cohomology of the de Rham-Witt complex in terms of coherent complexes over
the Raynaud ring. I will briefly mention a few complements (logarithmic
Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink
relative variants), and make a tentative list of open problems.
Monday (May 22), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. III.
Illusie will give his last talk on Thursday (May 25), 4:30 p.m, room E 206.
Here are the references to [BMS], i.e., to the following e-print:
https://arxiv.org/pdf/1602.03148.pdf
The statement that H^i(eta_fC) = H^i(C)/(Ker f : H^i(C) -> H^i(C)) is
[BMS, Lemma 6.4].
The statement that (eta_fC^{.})/f -> H^{.}(C^{.}/f) is compatible with
d on the lhs and Bockstein on the rhs, and is a quasi-isomorphism, is
[BMS, Prop. 6.12].
No more meetings this quarter.
The conference "Interactions between Representation Theory and Algebraic Geometry" will take place at the University of Chicago on August 21-25. Details can be found at https://sites.google.com/site/2017uchicagomathconference/ List of speakers: Dima Arinkin Roman Bezrukavnikov Bharghav Bhatt Alexander Braverman Helene Esnault Pavel Etingof Michael Finkelberg Jean-Marc Fontaine Dmitry Kaledin Mikhail Kapranov Masaki Kashiwara Ivan Losev George Lusztig Ivan Mirkovic Andrei Okounkov Raphael Rouquier Takeshi Saito Wolfgang Soergel Catharina Stroppel Eric Vasserot Xinwen Zhu
The conference "Interactions between Representation Theory and Algebraic Geometry" will take place at the University of Chicago next week (August 21-25). Details can be found at https://sites.google.com/site/2017uchicagomathconference/ https://sites.google.com/site/2017uchicagomathconference/schedule List of speakers: Dima Arinkin Roman Bezrukavnikov Bharghav Bhatt Alexander Braverman Helene Esnault Pavel Etingof Michael Finkelberg Jean-Marc Fontaine Dmitry Kaledin Mikhail Kapranov Masaki Kashiwara Ivan Losev George Lusztig Ivan Mirkovic Andrei Okounkov Raphael Rouquier Takeshi Saito Wolfgang Soergel Catharina Stroppel Eric Vasserot Xinwen Zhu
No seminar on Thursday (Oct 5). Nick Rozenblyum will continue on Oct 9 (Monday), 4:30 p.m, room E 206.
Nick Rozenblyum will continue on Monday (Oct 9), 4:30 p.m, room E 206.
No seminar on Thursday. Monday (Oct 16), 4:30 p.m, room E 206. Nick Rozenblyum. A "naive" approach to topological cyclic homology and the cyclotomic trace. III.
Monday (Oct 16), 4:30 p.m, room E 206. Nick Rozenblyum. A "naive" approach to topological cyclic homology and the cyclotomic trace. III.
No seminar on Thursday.
FRIDAY (Oct 20), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The Casselman-Jacquet functor and
Deligne-Lusztig duality for (g,K)-modules.
Abstract
We will discuss a recent joint work with T.-H. Chen and A. Yom Din in
which a new perspective on the Casselman-Jacquet functor is given. We
will also discuss the Serre functor on the category of (g,K)-modules and
its relation to the phenomenon of "miraculous duality" that recently
appeared in the context of the geometric Langlands theory.
FRIDAY (Oct 20), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The Casselman-Jacquet functor and
Deligne-Lusztig duality for (g,K)-modules.
Abstract
We will discuss a recent joint work with T.-H. Chen and A. Yom Din in
which a new perspective on the Casselman-Jacquet functor is given. We
will also discuss the Serre functor on the category of (g,K)-modules and
its relation to the phenomenon of "miraculous duality" that recently
appeared in the context of the geometric Langlands theory.
IF Sasha Beilinson gets well by Monday, he will explain some
Preliminaries for Bhatt's future talk.
(Sasha's talk is scheduled for Monday Oct 23, 4:30 p.m, room E 206).
Abstract
On November 13 Bhargav Bhatt will be giving a talk about a new approach to
the integral p-adic Hodge theory of Bhatt-Morrow-Scholze. The basic
classical structures that appear in the construction are lambda-rings and
(some of) Fontaine's rings. This is an introductory talk about them.
Beilinson's Monday talk is canceled (he is not quite well). So the next meeting is on Thursday Oct 26 (Nick Rozenblyum will finish his series of talks).
Thursday (Oct 26), 4:30 p.m, room E 206. Nick Rozenblyum. A "naive" approach to topological cyclic homology and the cyclotomic trace. IV.
Oct 30 (Monday), 4:30 p.m, room E 206.
Akhil Mathew. THH and crystalline cohomology. I.
Abstract
Given a commutative F_p-algebra, its topological Hochschild homology is
equipped with certain structures: a circle action, and a (cyclotomic)
Frobenius operator. Work of Bhatt-Morrow-Scholze shows that, for smooth
schemes in characteristic p, these structures recover crystalline
cohomology (an analogous variant over O_C recovers A_{inf}-cohomology).
The construction is based on a filtration on THH constructed via descent
to the regular semiperfect case. In these talks, I will explain the
details of the construction of this filtration and the relationship with
crystalline cohomology.
Thursday (Nov 2), 4:30 p.m, room E 206.
Akhil Mathew. THH and crystalline cohomology. II.
Abstract
Given a commutative F_p-algebra, its topological Hochschild homology is
equipped with certain structures: a circle action, and a (cyclotomic)
Frobenius operator. Work of Bhatt-Morrow-Scholze shows that, for smooth
schemes in characteristic p, these structures recover crystalline
cohomology (an analogous variant over O_C recovers A_{inf}-cohomology).
The construction is based on a filtration on THH constructed via descent
to the regular semiperfect case. In these talks, I will explain the
details of the construction of this filtration and the relationship with
crystalline cohomology.
Monday (Nov 6), 4:30 p.m, room E 206.
Alexander Beilinson. Preliminaries for Bhatt's November talk.
Abstract
On November 13 Bhargav Bhatt will be giving a talk about a new approach to
the integral p-adic Hodge theory of Bhatt-Morrow-Scholze. The basic
classical structures that appear in the construction are lambda-rings and
(some of) Fontaine's rings. This is an introductory talk about them.
No seminar on Thursday. Faltings is giving Albert lectures this week, see http://math.uchicago.edu/research/lecture-series/albert/ Next Monday (Nov 13) there will be a talk by Bhargav Bhatt, see http://math.uchicago.edu/research/calendar/
Monday (Nov 13), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan). Canonical deformations of de Rham
cohomology.
Abstract
In arithmetic geometry, there are multiple instances where the de Rham
cohomology of a smooth variety admits a canonical (and highly nontrivial)
deformation. Examples of such deformations include:
(a) crystalline cohomology over a perfect ring of characteristic p.
(b) the recently constructed A_{inf}-cohomology theory in p-adic Hodge
theory (jointly with Morrow and Scholze).
Conjecturally, there should be others, including:
(c) a ''q-deformation'' of de Rham cohomology (in the sense of Aomoto,
Jackson, Scholze) when working over an unramified base ring.
(d) a cohomological lift of the theory of Breuil-Kisin modules when
working over the ring of integers of a p-adic field.
In my talk, I will explain a general site-theoretic framework that
produces deformations of de Rham cohomology. This framework specializes to
(re)produce all the theories above. In the previously known cases (a) and
(b), the new construction is simpler and yields slightly finer
information. The key idea is to search for a deformation that carries a
Frobenius. (This is a report on a joint work in progress with Peter
Scholze.)
Thursday (Nov 16), 4:30 p.m, room E 206.
Victor Ginzburg. Differential operators on G/U and the Gelfand-Graev action.
Abstract
The variety G/U, where G is a semisimple group and U its maximal unipotent
subgroup, plays an important role in representation theory. In a joint
work with D. Kazhdan, we study the algebra D(G/U) of algebraic
differential operators on G/U, and its quasi-classical analogue: the
algebra of regular functions on the cotangent bundle. A long time ago,
Gelfand and Graev have constructed an action of the Weyl group on D(G/U)
by algebra automorphisms. The Gelfand-Graev construction was not
algebraic, it involved the Hilbert space L^2(G/U) in an essential way. I
will explain an algebraic construction of the Gelfand-Graev action as
well as its quasi-classical counterpart. Our approach is based on
Hamiltonian reduction and involves the ring of Whittaker differential
operators on G/U, a twisted analogue of D(G/U). If time permits, I'll
also discuss results of Laumon-Kazhdan and
Bezrukavnikov-Braverman-Positselskii concerning the category of
D(G/U)-modules and its etale analogue.
Monday (Nov 20), 4:30 p.m, room E 206.
Victor Ginzburg. Nil Hecke algebras and Whittaker D-modules.
Abstract
Given a complex semisimple group G, Kostant and Kumar defined a nil Hecke
algebra that may be viewed as a degenerate version of the double affine
nil Hecke algebra introduced by Cherednik. It turns out that there is an
isomorphism between the spherical subalgebra of the nil Hecke algebra with
a Whittaker type quantum Hamiltonian reduction of the algebra of
differential operators on G. This isomorphism provides an interesting
description of the category of Whittaker D-modules on G, considered by
Drinfeld, in terms of modules over the nil Hecke algebra. Our isomorphism
also has an interpretation in terms of geometric Satake and the Langlands
dual group. Specifically, it provides a bridge between very differently
looking descriptions of equivariant Borel-Moore homology of the affine
flag variety (due to Kostant and Kumar) and of the affine Grassmannian
(due to Bezrukavnikov and Finkelberg), respectively.
Happy Thanksgiving! The next meeting will be on Thursday November 30. Speaker: Gus Lonergan (a student of Bezrukavnikov). Title: Steenrod operations and the quantum Coulomb branch. The abstract can be found at http://math.uchicago.edu/research/calendar/
Thursday (Nov 30), 4:30 p.m, room E 206.
Gus Lonergan (MIT). Steenrod operations and the quantum Coulomb branch.
Abstract
It is a famous fact that the convolution algebra A of G(O)-equivariant
homology of Gr=G(K)/G(O) is commutative. However, its one-parameter
deformation A_h given by the G(O)\rtimes C*-equivariant homology of Gr is
certainly not commutative. Morally, the reason is as follows. Using
Beilinson-Drinfeld Grassmannians, one may present the convolution
multiplication of A in a manifestly commutative way, in terms of
specialization. Loop rotation C* also acts on the BD-Grassmannian (defined
over affine space), but it does not act trivially on the base, and so
C*-equivariance is incompatible with specialization.
What is do be done about this? We introduce a variant of the
BD-Grassmannian, where it is possible to approximate the action of C* by
an action of a finite subgroup which acts trivially on the base. The
resulting equivariant specialization map, in conjunction with Steenrod's
construction, provides a map from A (mod p) into the center of A_h (mod
p), which is a lift of the Frobenius endomorphism of A. Thus A_h is a
Frobenius-constant quantization. The theory of Frobenius-constant
quantizations has been developed by Bezrukavnikov-Kaledin in order to
transport "the characteristic p method" of (commutative) algebraic
geometry into the non-commutative setting.
The same idea may be used to prove Frobenius-constancy of the (mod p)
quantum Coulomb branch B_h of Braverman-Finkelberg-Nakajima. B_h is
defined in essentially the same way as A_h, i.e. as the convolution
algebra of G(O)\rtimes C*-equivariant Borel-Moore homology of a certain
sub-bundle R of a certain infinite-dimensional vector bundle T over Gr.
The situation here is more technical, since one has to work with schemes
of infinite type, dimension theories etc. Since these technicalities are
somewhat orthogonal to the main ideas of the talk, I prefer to explain the
situation for A_h first, and then - time permitting - indicate how it
generalizes to B_h.
No more meetings of the seminar this quarter.
No seminar this week.
The first meeting is *next* week on Thursday:
Jan 11 (Thursday), 4:30 p.m, room E 206.
Akhil Mathew. A gentle approach to the de Rham-Witt complex.
Abstract
The de Rham-Witt complex of a smooth algebra over a perfect field provides
a chain complex representative of its crystalline cohomology, a canonical
characteristic zero lift of its algebraic de Rham cohomology. We describe
a simple approach to the construction of the de Rham-Witt complex based on
the elementary notion of a "Dieudonn\'e complex." This relates to a
homological operation L\eta_p on the derived category, introduced by
Berthelot and Ogus. This is joint work with Bhargav Bhatt and Jacob Lurie.
Tomorrow (Thursday), 4:30 p.m, room E 206.
Akhil Mathew. A gentle approach to the de Rham-Witt complex.
Abstract
The de Rham-Witt complex of a smooth algebra over a perfect field provides
a chain complex representative of its crystalline cohomology, a canonical
characteristic zero lift of its algebraic de Rham cohomology. We describe
a simple approach to the construction of the de Rham-Witt complex based on
the elementary notion of a "Dieudonn\'e complex." This relates to a
homological operation L\eta_p on the derived category, introduced by
Berthelot and Ogus. This is joint work with Bhargav Bhatt and Jacob Lurie.
Jan 15 (Monday), 4:30 p.m, room E 206. Akhil Mathew will finish his talk on the derived de Rham-Witt complex.
A draft of the article on the de Rham-Witt complex (by Bhatt, Lurie, and Akhil Mathew) is here: http://math.uchicago.edu/~drinfeld/de_Rham-Witt.pdf As announced before, Akhil will finish his talk on the de Rham-Witt complex tomorrow (Monday), 4:30 p.m, room E 206.
Thursday (Jan 18), 4:30 p.m, room E 206.
Akhil Mathew. Kaledin's degeneration theorem and topological Hochschild
homology.
Abstract
For a smooth proper variety over a field of characteristic zero, the
Hodge-to-de Rham spectral sequence (relating the cohomology of
differential forms to de Rham cohomology) is well-known to degenerate, via
Hodge theory. A "noncommutative" version of this theorem has been proved
by Kaledin for smooth proper dg categories over a field of characteristic
zero, based on the technique of reduction mod p. Here differential forms
are replaced with Hochschild homology and de Rham cohomology with periodic
cyclic homology.
I will describe a short proof of Kaledin's theorem using the theory of
topological Hochschild homology, which provides a canonical one-parameter
deformation of Hochschild homology in characteristic p.
No seminar next week. ***** On Jan 29 (and probably on Feb 1) Alexander Petrov (Harvard) will speak on his work "The Gauss-Manin connection on the periodic cyclic homology" (joint with D.Vaintrob and V.Vologodsky), see https://arxiv.org/abs/1711.02802
Monday (Jan 29), 4:30 p.m, room E 206.
Alexander Petrov (Harvard). Periodic cyclic homology in positive
characteristics.
Abstract
This is a joint work with Dmitriy Vaintrob and Vadim Vologodsky. Periodic
cyclic homology is an invariant of a dg-category which recovers the
2-periodization of the de Rham cohomology of a smooth algebraic variety
when applied to its bounded derived category(if characteristics of the
base field is 0 or is bigger than the dimension). For a dg-category over a
smooth algebra, the periodic cyclic homology carries a flat connection. We
study these modules with connections in the case when the base field is of
positive characteristic and exhibit the relative Deligne-Illusie
degeneration theorem for it and an analog of Katz p-curvature theorem. The
main tools are Kaledin's conjugate filtration on the periodic cyclic
homology and the Tate cohomology. I also hope to discuss the relation of
these objects to the topological Hochschild homology and the cyclotomic
structure on it and, in particular, explain an algebraic construction of
the topological periodic cyclic homology for a dg-category over a perfect
field of positive characteristics.
Alexander Petrov will finish his talk on Thursday (Feb 1), 4:30 p.m, room E 206. A draft of the relevant unpublished work by Petrov and Vologodsky is here: http://math.uchicago.edu/~drinfeld/Petrov-Vologodsky.pdf
No seminar on Monday. On Thursday Sam Raskin will give a talk on Whittaker
models in classical representation theory (to be followed by a talk on
Whittaker models in geometric representation theory over local fields).
******
Feb 8 (Thursday), 4:30 p.m, room E 206.
Sam Raskin. Introduction to Whittaker models
Abstract
In this talk, we will introduce Whittaker models for representations of
p-adic reductive groups. One of the major goals of the talk is to explain
how to realize Whittaker models via compact open subgroups, and how these
methods help to perform some basic calculations. This talk will be quite
elementary.
Thursday (Feb 8), 4:30 p.m, room E 206.
Sam Raskin. Introduction to Whittaker models
Abstract
In this talk, we will introduce Whittaker models for representations of
p-adic reductive groups. One of the major goals of the talk is to explain
how to realize Whittaker models via compact open subgroups, and how these
methods help to perform some basic calculations. This talk will be quite
elementary.
Monday (Feb 12), 4:30 p.m, room E 206.
Sam Raskin. Calculation of some Hecke-Whittaker algebras.
Abstract
This talk has two goals. First, we will explain how to adapt the ideas
from the previous talk into the setting of geometric representation
theory. Second, we will explain the calculation of certain Hecke algebras
arising naturally from the compact approximation considerations from the
first talk. The latter results work in parallel in the classical setting
as in the geometric setting, and may viewed as a variant of Ginzburg's
talk from last semester.
An application of these methods will be given in a third talk, whose
subject will be Frenkel-Gaitsgory's conjecture on Beilinson-Bernstein
localization for the affine Grassmannian.
Sam Raskin's notes related to his yesterday talk are here:
http://math.uchicago.edu/~drinfeld/Raskin_on_Heisenberg.pdf
Sam told me that there were many talks on the notion of group
acting on a category and so on at a seminar held in Paris last month, and
many notes are available here:
<https://sites.google.com/site/winterlanglands2018/notes-of-talks>. He
says it's a good place to go look if people want to learn more.
********
No seminar on Thursday.
Sam Raskin will give his last talk on Monday (Feb 19).
Title: Affine Beilinson-Bernstein at the critical level for GL_2
Abstract
There has long been interest in Beilinson-Bernstein localization
for the affine Grassmannian (or affine flag variety). First,
Kashiwara-Tanisaki treated the so-called negative level case in the 90's.
Some ten years later, Frenkel-Gaitsgory (following work of
Beilinson-Drinfeld and Feigin-Frenkel) formulated a conjecture at the
critical level and made some progress on it. Their conjecture is more
subtle than its negative level counterpart, but also more satisfying.
We will review the necessary background from representation theory of
Kac-Moody algebras at critical level, formulate the Frenkel-Gaitsgory
conjecture, and explain how to prove it for GL_2. Given the previous talks,
the proof will be more straightforward than the formulation.
Monday (Feb 19), 4:30 p.m, room E 206.
Sam Raskin. Affine Beilinson-Bernstein at the critical level for GL_2
Abstract
There has long been interest in Beilinson-Bernstein localization for the
affine Grassmannian (or affine flag variety). First, Kashiwara-Tanisaki
treated the so-called negative level case in the 90's. Some ten years
later, Frenkel-Gaitsgory (following work of Beilinson-Drinfeld and
Feigin-Frenkel) formulated a conjecture at the critical level and made
some progress on it. Their conjecture is more subtle than its negative
level counterpart, but also more satisfying.
We will review the necessary background from representation theory of
Kac-Moody algebras at critical level, formulate the Frenkel-Gaitsgory
conjecture, and explain how to prove it for GL_2. Given the previous
talks, the proof will be more straightforward than the formulation.
Dear All, Is there anybody who would volunteer to give in spring a seminar talk on the recent article https://arxiv.org/pdf/1711.06436 by Esnault and Groechenig? They prove an interesting particular case of an old conjecture of Carlos Simpson. The conjecture is about local systems on complex algebraic varieties, but the proof is number-theoretic (in some sense). The proof is simple (modulo some facts used as a "black box"). The article has only 13 pages; moreover, the first version of the article (which treats the projective case) has only 5 pages. In the talk I suggest to focus on the projective case. I think this article is beautiful and instructive. In particular, it is instructive to understand why the authors didn't prove Simpson's conjecture in full generality (instead of rigidity of the local system they have to assume a stronger property of "cohomological rigidity"). And who knows, maybe their method can be modified to prove Simpson's conjecture in full generality?
No meetings until March 5. On March 5 Nate Harman will speak on tensor categories.
March 5 (Monday), 4:30 p.m, room E 206.
Nate Harman. Dimensions and growth in tensor categories.
Abstract
The first half of the talk will be a general overview of the theory of
tensor categories. The focus will be on discussing various extra
structures and properties often imposed on tensor categories (rigidity,
pivotal structures, braidings, etc.) and giving plenty of examples of such
categories. In the second half we will focus on notions of dimension and
growth of objects in tensor categories, and discuss some recent results
and conjectures about them.
March 8 (Thursday), 4:30 p.m, room E 206. Nate Harman. Dimensions and growth in tensor categories. II.
No more meetings of the seminar in the winter quarter.
First meeting: April 5 (Thursday). ******* Here is a preliminary schedule. April 5: Roman Bezrukavnikov. April 9 and 12: Daniil Rudenko Apil 20 (Friday!) and April 23: Dennis Gaitsgory April 30: Zhiyuan Ding's talk on e-print arxiv.org/pdf/1711.06436 (by Esnault and Groechenig). May 3 and/or May 7: Ngo Bao Chau or Tsao-Hsien Chen will speak on their joint work. May 14: Dima Arinkin.
Thursday (April 5), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT). Characters and almost characters for p-adic
groups.
Abstract
The talk will be based on joint work (partly in progress) with Kazhdan and
Varshavsky. Characters of the finite group GL(n,F_q) are well know to
arise as the trace of Frobenius function of an irreducible perverse sheaf
on the algebraic group GL(n). I will present generalizations of this
involving unipotent characters and other invariant distributions (more
precisely, elements in the Bernstein center) on the group GL(F_q((t)) ).
Time permitting, I will discuss a possible approach to generalizing this
to other reductive groups.
Monday (April 9), 4:30 p.m, room E 206.
Daniil Rudenko. Depth conjecture for polylogarithms.
Abstract
Classical polylogarithms have been studied extensively since pioneering
works of Euler and Abel. It is known that they satisfy lots of functional
equations, but in weight >4 these equations are not known yet. Even in
weight 4 they were first found using heavy computer-assisted computations.
The main goal of the talk is to explain a conceptual proof of the depth
conjecture in weight 4, leading to the functional equations. Our approach
uses some new technics, including secondary polytopes, cluster algebras
and motivic correlators. To provide motivation we will also discuss the
conjectural motivic framework, explaining some properties of
polylogarithms, and the relation to Zagier conjecture.
The talk is based on joint work with A. Goncharov.
Thursday (April 12), 4:30 p.m, room E 206. Daniil Rudenko. Depth conjecture for polylogarithms. II.
Daniil Rudenko will continue on Monday (Apr 16) at 5 p.m. (yes, FIVE p.m !) in the usual room E206. Reason for the unusual time: the Namboodiri lecture, which is supposed to finish at 5 p.m. (but in practice, it could finish at 5 pm + epsilon).
FRIDAY (April 20), 4:30 p.m, room E 202.
Dennis Gaitsgory (Harvard). Quantum geometric Langlands.
(Please notice the unusual day and room!)
Abstract
We will formulate the global quantum geometric Langlands conjecture, along
with the local-to-global compatibilities it is supposed to satisfy, and
with the emphasis on the dependence on the sign of the level.
Reminder: tomorrow (Friday) at 4:30 p.m. we have a talk
in an unusual room E 202.
Dennis Gaitsgory (Harvard). Quantum geometric Langlands.I.
Abstract
We will formulate the global quantum geometric Langlands conjecture, along
with the local-to-global compatibilities it is supposed to satisfy, and
with the emphasis on the dependence on the sign of the level.
Dennis Gaitsgory will give his second talk on Quantum Langlands on Monday (April 23), 4:30 p.m, room E 206. The materials of the Paris conference are here: https://sites.google.com/site/winterlanglands2018/notes-of-talks
Dennis privately told me that his today's lecture corresponds (more or less) to Sections 1.5 and 5.3 of the following file: http://www.iecl.univ-lorraine.fr/~Sergey.Lysenko/program_1.pdf This file and this information could be useful for you.
No seminar on Thursday, April 26. Presumably, on Monday April 30 there will be a meeting of the seminar with a talk about the work https://arxiv.org/abs/1711.06436 (In that work Esnault and Groechenig prove a special case of Carlos Simpson's conjecture on rigid local systems.)
Monday (April 30), 4:30 p.m, room E 206.
Zhiyuan Ding. On the work on cohomologically rigid local systems by
Esnault and Groechenig.
Abstract
Let X be a smooth complex projective variety and \rho an irreducible
representation of \pi_1(X) whose determinant has finite order. \rho is
said to be rigid (resp. cohomologically rigid) if it has no nontrivial
deformations (resp. first order deformations) preserving the determinant.
Cohomological rigidity implies rigidity.
Carlos Simpson conjectured that rigidity implies integrality. Esnault and
Groechenig proved that cohomological rigidity implies integrality, see
https://arxiv.org/abs/1711.06436
I will explain their proof, which is based on reduction of the variety
modulo p.
No seminar on Thursday May 3. On Monday May 7 Ngo Bao Chau will speak on his article "On the Hitchin fibration for algebraic surfaces" (joint with Tsao-Hsien Chen), see https://arxiv.org/pdf/1711.02592.
Monday (May 7), 4:30 p.m, room E 206.
Ngo Bao Chau. On the Hitchin fibration for algebraic surfaces.
Abstract
Simpson constructs the Hitchin map from the moduli stack of Higgs bundles
over an an arbitrary smooth algebraic varieties X to a vector space,
generalizing Hitchin’s construction in the one-dimensional case. In
one-dimensional case we understand well the geometry of the Hitchin
fibration as opposed to the higher-dimensional case where very little is
known. I will report on a joint work with Chen in which we start to
investigate the two-dimensional case.
No seminar on Thursday May 10. On Monday May 14 Dima Arinkin will speak on the classical limit of the (local) geometric Langlands correspondence.
Monday (May 14), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Classical limit of the (local)
geometric Langlands correspondence.
Abstract
The classical limit of the global geometric Langlands correspondence is
the conjectural Fourier-Mukai equivalence between the Hitchin fibrations
for a reductive group G and its dual. While there was a significant
progress on this statement for G=GL(n), much less is known about the case
of general G.
In my talk, I plan to review the global setting, and then focus on the
classical limit of the _local_ geometric Langlands correspondence. My goal
is to explain new techniques and ideas that are available in the local
case.
Dima Arinkin will continue on Thursday (May 17), 4:30 p.m, room E 206. > Classical limit of the local geometric Langlands correspondence. > > Abstract > > The classical limit of the global geometric Langlands correspondence is > the conjectural Fourier-Mukai equivalence between the Hitchin fibrations > for a reductive group G and its dual. While there was a significant > progress on this statement for G=GL(n), much less is known about the case > of general G. > > In my talk, I plan to review the global setting, and then focus on the > classical limit of the _local_ geometric Langlands correspondence. My goal > is to explain new techniques and ideas that are available in the local > case.
No more meetings of the seminar this quarter.
As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. We begin on October 8 with A.Braverman’s talk, whose title is "Topologically twisted 3-dimensional gauge theories and local Langlands duality". After that, there will be talks on Oct 11 (by Semen Gindikin) and on Oct 18 (by Yotam Hendel, a student of J.Bernstein and Rami Aizenbud).
Monday (Oct 8), 4:30 p.m, room E 206.
Alexander Braverman (Toronto). Topologically twisted 3-dimensional gauge
theories and local Langlands duality.
Abstract
I will start the talk by presenting a series of rather surprising
(mathematical) conjectures involving various equivalences of categories;
some of these conjectures provide new understanding of local geometric
Langlands duality for the group GL(n). In the main body of the talk I
would like to explain how one can "invent" these conjectures while
studying super-symmetric 3-dimensional quantum field theories (the
relevant background will be explained in the talk, no familiarity with
quantum field theory will be assumed). If time permits, I will explain
some mathematical evidence for these conjectures (mostly in the case of
GL(2)).
P.S. As far as I understand, Braverman's Monday talk will be related to the following article: https://arxiv.org/pdf/1807.09038.pdf > Monday (Oct 8), 4:30 p.m, room E 206. > Alexander Braverman (Toronto). Topologically twisted 3-dimensional gauge > theories and local Langlands duality. > > > Abstract > > I will start the talk by presenting a series of rather surprising > (mathematical) conjectures involving various equivalences of categories; > some of these conjectures provide new understanding of local geometric > Langlands duality for the group GL(n). In the main body of the talk I > would like to explain how one can "invent" these conjectures while > studying super-symmetric 3-dimensional quantum field theories (the > relevant background will be explained in the talk, no familiarity with > quantum field theory will be assumed). If time permits, I will explain > some mathematical evidence for these conjectures (mostly in the case of > GL(2)). > > >
Thursday (Oct 11), 4:30 p.m, room E 206.
Semen Gindikin (Rutgers). Horospheric transform on symmetric spaces as a
curved Radon transform.
Abstract
Almost exactly 60 years ago, Gelfand remarked that Harmonic Analysis on
symmetric spaces has a geometrical twin - the horospherical transform,
similarly to the usual Fouier integral having as a twin the Radon
transform. His plan was to reconstruct the theory of representations as a
theory of the horospherical transform. However, this program was realized
only in some fragments.
I will talk about some new progress, using the example of the inversion of
the horospherical transform on Riemannian symmetric spaces. It turns out
that this problem is in a sense trivial: it is equivalent to the similar
problem for the flat model, which is solved by the Abelian Fourier
transform.
No seminar on Monday Oct 15.
Thursday (Oct 18), 4:30 p.m, room E 206.
Yotam Hendel (Weizmann Institute). Singularity properties of convolutions
of algebraic morphisms and applications.
Abstract
In analysis, the convolution of two functions results in a smoother,
better behaved function. It is interesting to ask whether there exists a
geometric analogue of this phenomenon.
Let f and g be two morphisms from algebraic varieties X and Y to an
algebraic group G. We define their convolution to be the morphism f*g from
X x Y to G obtained by first applying each morphism and then multiplying
using the group structure of G.
In this talk, we present some properties of this convolution operation, as
well as a recent result which states that after finitely many self
convolutions every dominant morphism f:X->G from a smooth, absolutely
irreducible variety X to an algebraic group G becomes flat with reduced
fibers of rational singularities (this property is abbreviated FRS).
The FRS property is of particular interest since by works of Aizenbud and
Avni, FRS morphisms are characterized by having fibers whose point count
over the finite rings Z/p^kZ is well-behaved. This leads to applications
in probability, group theory, representation growth and more.
We will discuss some of these applications, and the main ideas of the
proof which utilize model-theoretic methods.
This is joint work with Itay Glazer.
Thursday (Oct 18), 4:30 p.m, room E 206.
Yotam Hendel (Weizmann Institute). Singularity properties of convolutions
of algebraic morphisms and applications.
Abstract
In analysis, the convolution of two functions results in a smoother,
better behaved function. It is interesting to ask whether there exists a
geometric analogue of this phenomenon.
Let f and g be two morphisms from algebraic varieties X and Y to an
algebraic group G. We define their convolution to be the morphism f*g from
X x Y to G obtained by first applying each morphism and then multiplying
using the group structure of G.
In this talk, we present some properties of this convolution operation, as
well as a recent result which states that after finitely many self
convolutions every dominant morphism f:X->G from a smooth, absolutely
irreducible variety X to an algebraic group G becomes flat with reduced
fibers of rational singularities (this property is abbreviated FRS).
The FRS property is of particular interest since by works of Aizenbud and
Avni, FRS morphisms are characterized by having fibers whose point count
over the finite rings Z/p^kZ is well-behaved. This leads to applications
in probability, group theory, representation growth and more.
We will discuss some of these applications, and the main ideas of the
proof which utilize model-theoretic methods.
This is joint work with Itay Glazer.
The slides of Yotam Hendel's talk are here:
http://math.uchicago.edu/~drinfeld/Hendel's_talk.pdf
******
Monday (Oct 22), 4:30 p.m, room E 206.
Dimitri Wyss (Jussieu). p-adic integration and geometric stabilization.
Abstract
I will explain a new proof of the geometric stabilization theorem for
Hitchin fibers, a key ingredients in Ngo's proof of the fundamental lemma.
Our approach relies on ideas of Denef-Loeser and Batyrev on p-adic
integration as well as the classical limit of the geometric Langlands
correspondence. This is joint work with Michael Groechenig and Paul
Ziegler.
No seminar on Thursday (Oct 25).
No seminar on Monday (Oct 29). On Thursday (Nov 1) Dmitry Kaledin will speak on Hochschild-Witt homology. ***** The slides of Gindikin's talk are here: http://math.uchicago.edu/~drinfeld/Gindikin's_talk.pdf
Thursday (Nov 1), 4:30 p.m, room E 206.
Dmitry Kaledin (Moscow). Hochschild-Witt Homology.
Abstract
Hochschild-Witt Homology is a homology theory for pairs of an associative
algebra A over a finite field, and an A-bimodule M. It generalizes the
usual Hochschild Homology in exactly the same way as the de Rham-Witt
complex generalizes the de Rham complex, and gives a full non-commutative
generalization of crystalline cohomology for varieties over a finite
field. I will give a general overview of the subject, and show how it
clarifies the classical theory of the de Rham-Witt complex.
No seminar on Monday (Nov 5).
*********
On Thursday (Nov 8) Akhil Mathew will speak on the arc-topology.
Abstract
I will discuss a Grothendieck topology on the category of quasi-compact
quasi-separated schemes called the "arc-topology”. Covers in the
arc-topology are tested via rank \leq 1 valuation rings. This topology is
motivated by classical questions in algebraic K-theory. Our main result is
that etale cohomology with torsion coefficients satisfies arc-descent.
Using these tools, I will describe an application to Artin-Grothendieck
vanishing in rigid analytic geometry, which strengthens results of Hansen.
This is joint work with Bhargav Bhatt.
Thursday (Nov 8), 4:30 p.m, room E 206.
Akhil Mathew. The arc-topology.
Abstract
I will discuss a Grothendieck topology on the category of quasi-compact
quasi-separated schemes called the "arc-topology”. Covers in the
arc-topology are tested via rank \leq 1 valuation rings. This topology is
motivated by classical questions in algebraic K-theory. Our main result is
that etale cohomology with torsion coefficients satisfies arc-descent.
Using these tools, I will describe an application to Artin-Grothendieck
vanishing in rigid analytic geometry, which strengthens results of Hansen.
This is joint work with Bhargav Bhatt.
No seminar this week. On Nov 19 Joel Kamnitzer (Toronto) will speak on symplectic duality.
Monday (Nov 19), 4:30 p.m, room E 206.
Joel Kamnitzer (Toronto). Overview of symplectic duality.
Abstract
Symplectic duality is a collection of relationships between pairs of
conical symplectic singularities and their quantizations. I will discuss
these relationships, which include theorems and conjectures of
Braden-Licata-Proudfoot-Webster, Hikita, and others. I will also discuss
many examples of symplectic dual pairs, including hypertoric varieties,
affine Grassmannian slices / quiver varieties, and the
Braverman-Finkelberg-Nakajima construction.
No more seminars this quarter.
Dear All,
Is there anybody who would volunteer to give in winter a seminar talk (or
maybe talks) on the following subject:
delta-rings and Witt vectors.
Roughly, one needs an expanded version of the first half of the following
talk by Beilinson:
http://math.uchicago.edu/~drinfeld/Seminar-2019/Sasha-2017.pdf
(these are my notes, which are far from being perfect).
This includes:
a) some standard material on Witt vectors,
b) some material from Bhatt's Lecture 2, see
http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/lecture2-delta-rings.pdf
c) Joyal's 1985 article "delta-anneaux et vecteurs de Witt", see
http://math.uchicago.edu/~drinfeld/Seminar-2019/Joyal-1985.pdf
The material from Joyal's article is also contained in pages 1-19 of
Borger's lecture notes, see
https://maths-people.anu.edu.au/~borger/classes/copenhagen-2016/LectureNotes.pdf
https://maths-people.anu.edu.au/~borger/classes/copenhagen-2016/index.html
If somebody wants to talk with Beilinson or me before making a decision to
volunteer, please feel free to contact us.
If you decide to volunteer then it would makes sense to have a meeting with
Beilinson and/or me and then to prepare the talk(s) in contact with one of
us.
In winter we plan to devote most of the time to studying some material related to Scholze's ICM talk https://arxiv.org/pdf/1712.03708.pdf especially prismatic cohomology. This is a new p-adic cohomology theory for p-adic (formal) schemes developed by Bhatt and Scholze, which is probably "the right one". We hope that many people (algebraic geometers, number theorists, …) would benefit from studying this theory. The good news is that Bhatt's lecture notes on prismatic cohomology are locally understandable, see http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/ Another good news is that for schemes over F_p prismatic cohomology amounts to crystalline cohomology. Moreover, it seems that once we understand crystalline cohomology in a certain way, it will be possible for us to understand prismatic cohomology for arbitrary p-adic formal schemes. So the seminar will probably start with an exposition of crystalline cohomology from scratch (by Brian Lawrence). I hope that students will be able to understand most of the material (of course, this will require some work). The theory of prismatic cohomology is based on the elementary notion of delta-ring (roughly, a ring equipped with a lift of Frobenius). We need a volunteer who would give talk(s) explaining basic facts about delta-rings and their relation to Witt vectors. I am sending a separate message about this subject and the relevant references.
The seminar begins with two talks by Bhargav Bhatt on Monday Jan 7 and Thursday Jan 10. He will give an overview of prismatic cohomology. This is a new p-adic cohomology theory for p-adic (formal) schemes developed by Bhatt and Scholze, which is probably "the right one", see http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/ Most of the time in winter will be devoted to a detailed study of the theory.
Monday (Jan 7), 4:30 p.m, room E 206.
Bhargav Bhatt (University of Michigan) will begin his overview of
prismatic cohomology.
Abstract
Prismatic cohomology is a cohomology theory for p-adic formal schemes. For
schemes of characteristic p, it yields (a Frobenius descent of)
crystalline cohomology. When working over the ring of integers of a local
field, this theory provides a mechanism to control the p-torsion in the
etale cohomology of the generic fibre in terms of the crystalline
cohomology of the special fibre.
In these lectures, I'll give an overview of the construction of this
cohomology theory and explain why (in some situations) it can be computed
as a "q-deformation" of the de Rham complex.
Bhargav Bhatt will give his second talk on Thursday (Jan 10), 4:30 p.m, room E 206. The name "prismatic cohomology" goes back (in part) to the picture used by the rock band Pink Floyd, see https://en.wikipedia.org/wiki/The_Dark_Side_of_the_Moon
No seminar on Monday.
Thursday (Jan 17), 4:30 p.m, room E 206.
Brian Lawrence. Introduction to Crystalline Cohomology. I
Abstract
I'll give an introduction to crystalline cohomology, assuming no
background beyond familiarity with schemes. In this first talk, I'll talk
about the characteristic zero case, and explain how Grothendieck
interpreted vector bundles with an integrable connection as sheaves on a
certain site.
Thursday (Jan 17), 4:30 p.m, room E 206.
Brian Lawrence. Introduction to Crystalline Cohomology. I
Abstract
I'll give an introduction to crystalline cohomology, assuming no
background beyond familiarity with schemes. In this first talk, I'll talk
about the characteristic zero case, and explain how Grothendieck
interpreted vector bundles with an integrable connection as sheaves on a
certain site.
No seminar on Monday (Jan 21). On Thursday Jan 24 there will be a talk by Dennis Gaitsgory. Title: The fundamental local equivalence in quantum geometric Langlands. ***************** Here are references related to today's talk. 1. The book by Berthelot and Ogus www.math.hawaii.edu/~pavel/cmi/References/Berthelot_Notes_on_Crystalline_Cohomology.pdf Here the first theorem, relating connections on an O_X-module to a certain isomorphism on the thickened diagonal P, is Proposition 2.9. The second theorem, relating integrable connections on an O_X-module to "stratifications" is Proposition 2.11. (It must be understood that a ring homomorphism from "differential operators on O_X" to "differential operators on E" is the same as a connection on E.) 2. Grothendieck's article "Crystals and the De Rham Cohomology os Schemes", available at https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Alltenlectures.pdf On pp. 331 and 332 are definitions of the "infinitesimal" and the "stratifying" topos; in our situation (where X is smooth) they are the same. The theorem that a module with stratification is the same as a crystal on the stratifying topos is proven on pp. 332-333. (Grothendieck calls crystals "special Modules" here.)
Thursday (Jan 24), 4:30 p.m, room E 206.
Dennis Gaitsgory (Harvard). The fundamental local equivalence in quantum
geometric Langlands.
Abstract
We will outline the construction of the equivalence between twisted
(=metaplectic) Whittaker sheaves on the affine Grassmannian and modules
over the Langlands dual quantum group.
Monday (Jan 28), 4:30 p.m, room E 206. Brian Lawrence will give his second talk on crystalline cohomology.
Thursday (Jan 31), 4:30 p.m, room E 206. Brian Lawrence will give his third talk on crystalline cohomology.
Monday (Feb 4), 4:30 p.m, room E 206.
Alexander Beilinson. The comparison between crystalline and de Rham
cohomology.
Abstract
I will explain a theorem of Berthelot that identifies crystalline
cohomology of a variety in characteristic p with de Rham cohomology of its
lifting in characteristic zero.
Beilinson will continue on Thursday Feb 7 (at 4:30 p.m, in room E 206).
No seminar on Monday February 11.
Notes of Sasha's talks are here: http://math.uchicago.edu/~drinfeld/Seminar-2019/Sasha's_notes.pdf
Thursday (February 14), 4:30 p.m, room E 206.
Alexander Beilinson. A stacky approach to crystalline cohomology.
Abstract
I will explain how crystalline cohomology can be seen as cohomology of
some natural stack. (In later talks we will use a similar stacky approach
to prismatic cohomology.)
Monday (February 18), 4:30 p.m, room E 206.
V.Drinfeld. The prismatization of the affine line over F_p.
Abstract
In his talk Beilinson associated to an F_p-scheme X a p-adic stack called
the crystallization of X. I will prove that if X is the affine line this
stack identifies with the cone of multiplication by p in the scheme of
Witt vectors. (In my next talk I will prove a similar result for any
F_p-scheme X.)
The seminar on Monday (February 18) is CANCELED (to avoid conflict with Matsuki's lecture). Our next meeting is on Thursday February 21. (I will speak on the prismatization of the affine line over F_p.) *********** The basic material on Witt vectors can be found in http://math.uchicago.edu/~drinfeld/Seminar-2019/Witt_vectors/Lang%20on%20Witt%20vectors.pdf http://math.uchicago.edu/~drinfeld/Seminar-2019/Witt_vectors/Lenstra%20on%20Witt%20vectors.pdf http://math.uchicago.edu/~drinfeld/Seminar-2019/Witt_vectors/Hesselholt%20on%20Witt%20vectors.pdf The ring *scheme* of Witt vectors is discussed in Lecture 26 of D.Mumford's "Lectures on curves on an algebraic surface".
Thursday (Feb 21), 4:30 p.m, room E 206.
V.Drinfeld. The prismatization of the affine line over F_p.
Abstract
In his talk Beilinson associated to an F_p-scheme X a p-adic stack called
the crystallization of X. I will prove that if X is the affine line this
stack identifies with the cone of multiplication by p in the scheme of
Witt vectors. (In my next talk I will prove a similar result for any
F_p-scheme X.)
Thursday (Feb 28), 4:30 p.m, room E 206.
Vladimir Drinfeld. The prismatization of a scheme of characteristic p.
Abstract
I will define the prismatization of a scheme X of characteristic p by
specifying its ring-valued points. Then I will identify the prismatization
of X with its crystallization (the latter was defined by Beilinson).
No seminar on Monday.
****************
Thursday (Feb 28), 4:30 p.m, room E 206.
Vladimir Drinfeld. The prismatization of a scheme of characteristic p.
Abstract
I will define the prismatization of a scheme X of characteristic p by
specifying its ring-valued points. Then I will identify the prismatization
of X with its crystallization (the latter was defined by Beilinson).
Thursday (Feb 28), 4:30 p.m, room E 206.
Vladimir Drinfeld. The prismatization of a scheme of characteristic p.
Abstract
I will define the prismatization of a scheme X of characteristic p by
specifying its ring-valued points. Then I will identify the prismatization
of X with its crystallization (the latter was defined by Beilinson).
Monday (March 4), 4:30 p.m, room E 206.
Vladimir Drinfeld. Prismatization. II.
Abstract
For an F_p-scheme X, I will identify the prismatization of X with its
crystallization.
If time permits, I will also say a few words about the prismatization of
p-adic formal schemes. (Details will be discussed in spring.)
No seminar on Thursday this week. No seminar on Monday and Thursday next week. The next meeting (the last one in winter) will be on FRIDAY NEXT WEEK at 4:10 p.m (room E 206). Speaker: Peter Scholze.
Please notice the *unusual date, time, and room* of the next meeting!
Friday (March 15), 4:10 p.m, room E 202.
Peter Scholze (Bonn University and MPI). The etale comparison for
prismatic cohomology.
Abstract
One can recover etale cohomology as Frobenius fixed points on prismatic
cohomology. We will explain the statement and proof of this result, and
how it relates to the tilting equivalence for etale cohomology of
perfectoid spaces.
No more meetings this quarter.
The first meeting of the seminar is on April 8 (i.e., the second Monday). Anthony Wang will explain a theorem of Joyal, which is very important for understanding prismatic cohomology. The theorem says that the forgetful functor from delta-rings to rings has a right adjoint, which is nothing else but the Witt vector functor.
Monday (April 8), 4:30 p.m, room E 206.
Anthony Wang. Delta-rings and Witt vectors.
Abstract
Delta-rings were introduced by Joyal to give an alternate approach to the
theory of Witt vectors. In his approach, the Witt vector functor is
realized as a right adjoint to the forgetful functor from delta-rings to
rings. In my talk, I explain why this right adjoint exists, and show how
Joyal's method recovers the more traditional method of defining Witt
vectors.
No seminar on Thursday (April 11). Next week Cong Xue (Cambridge) will speak on Monday and Thursday (April 15, 18). Title of her talks: Cohomologies of stacks of shtukas. *********** The notes of Borger's course on Witt vectors are available at https://maths-people.anu.edu.au/%7Eborger/classes/copenhagen-2016/LectureNotes.pdf *********** Exercise: using Joyal's description of W, construct an isomorphism between Ker (F:W\to W) and the divided powers additive group. (Here the base ring is the localization of Z at p). Note that this isomorphism played a central talk in my winter talks, and the construction was pretty long. Joyal's description of W allows one to give a short proof. The idea is as follows. Joyal describes the coordinate ring of W as the free delta-ring on 1 generator; he also describes in these terms the ring structure on W and the map F:W\to W. Using this description of W, it is straightforward to give an explicit description of Ker (F:W\to W).
Monday (April 15), 4:30 p.m, room E 206.
Cong Xue (Cambridge University). Cohomologies of stacks of shtukas.I.
Abstract
Let G be a connected split reductive group over a finite field F_q and X a
smooth projective geometrically connected curve over F_q. In the first
talk, I will recall the definition of stacks of G-shtukas and their l-adic
cohomology groups, which generalize the space of automorphic forms with
compact support over the function field of X. I will also
construct constant term morphisms on the cohomology groups.
In the second talk (on Thursday April 18), I will use the constant term
morphisms to show that the cohomology groups of stacks of shtukas are of
finite type as modules over the Hecke algebra at an unramified place. This
allows us to extend the excursion operators of V. Lafforgue from the space
of cuspidal automorphic forms to the space of all automorphic forms with
compact support, and gives the Langlands parametrization for some quotient
spaces of the latter, in the way compatible with parabolic induction.
Thursday (April 18), 4:30 p.m, room E 206., 4:30 p.m, room E 206. Cong Xue (Cambridge University). Cohomologies of the stacks of shtukas.II.
Cong Xue's notes of her second talk are at http://math.uchicago.edu/~drinfeld/Seminar-2019/Cong%20Xue_April%2018.pdf Cong Xue's summary of her results (in French) is at http://math.uchicago.edu/~drinfeld/Seminar-2019/Cong%20Xue-resume.pdf ******** No seminar on Monday (Apr 22). On Thursday (Apr 25) I will give my first talk on prismatization of mixed characteristic schemes.
Thursday (Apr 25), 4:30 p.m, room E 206.
Vladimir Drinfeld. The prismatization of the formal spectrum of Z_p .
Abstract
This is the first talk on prismatization of mixed characteristic schemes.
First, I will say a few words about the format of prismatic theory and fix
some terminology related to algebraic and formal stacks. Then I will
recall the definition of the prismatization of Spf Z_p given by me in
March and start exploring this stack.
Monday (Apr 29), 4:30 p.m, room E 206.
Vladimir Drinfeld. The prismatization of the formal spectrum of Z_p . II.
Abstract
We will continue exploring the stack \Sigma (i.e., the prismatization of
Spf Z_p). In particular, I will define a certain F-crystal on \Sigma,
which is the Breuil-Kisin analog of Z_p(-1).
Thursday (May 2), 4:30 p.m, room E 206.
Matthew Emerton. Breuil-Kisin modules: introduction and motivation.
Abstract
In this talk I will give an introduction to the theory of Breuil-Kisin
modules, starting from the original point of view adopted by Breuil and
Kisin, and explaining the connection with other ideas in the theory of
Galois representations for p-adic fields (such as Fontaine's p-adic Hodge
theory).
If time permits, I will then explain the relationship to more current
developments, such as A_{inf} cohomology and prismatic cohomology (and so
potentially make contact as well with ideas introduced in Drinfeld's
recent lectures).
Today's meeting of the seminar is CANCELED (the speaker is sick). ************ The notes of my talks in spring are available at http://math.uchicago.edu/~drinfeld/Seminar-2019/Spring/ (The file "q-de Rham prism.pdf" corresponds to the material I had no time to explain.) The notes of winter talks are at http://math.uchicago.edu/~drinfeld/Seminar-2019/Winter/
Monday (May 6), 4:30 p.m, room E 206. Depending on the situation, one of the following will happen: either Matt Emerton will speak on Breuil-Kisin modules or I will speak on prismatization of arbitrary p-adic schemes. I hope to be able to say something more definite on Sunday.
Monday (May 6), 4:30 p.m, room E 206.
Matthew Emerton. Breuil-Kisin modules: introduction and motivation.
Abstract
In this talk I will give an introduction to the theory of Breuil-Kisin
modules, starting from the original point of view adopted by Breuil and
Kisin, and explaining the connection with other ideas in the theory of
Galois representations for p-adic fields (such as Fontaine's p-adic Hodge
theory).
If time permits, I will then explain the relationship to more current
developments, such as A_{inf} cohomology and prismatic cohomology (and so
potentially make contact as well with ideas introduced in Drinfeld's
recent lectures).
Thursday (May 9), 4:30 p.m, room E 206. V.Drinfeld. Prismatization of arbitrary p-adic schemes. I will define the prismatization of an arbitrary p-adic scheme. I will also compute the prismatization of Spec (Z/p^n). For n=1 the answer is Spf Z_p . On the other hand, for n=\infty we get the prismatization of Spf Z_p, which is the stack \Sigma. I will explain what happens between n=1 and n=\infty (I was unable to guess the answer until I computed it.)
1. Attached is the file with Akhil's proofs of some basic facts. Lemma 2 of this file is the statement that I failed to prove during my talk. Lemma 3 of the file was formulated in my talk as a "fact" (without a proof). 2. During the first hour of my talk I defined "classical prismatization", then I explained why this functor cannot commute with projective limits. This explanation was not quite correct: I forgot that the functor lands in the category of stacks over \Sigma (rather than merely stacks). So my argument doesn't really show that if Y is the prismatization of X then Y is a space. It only proves that the *fibers* of the morphism Y\to\Sigma are spaces. To get a contradiction one cannot take X=Spf Z_p , but it is enough to take X to be the affine line over Z_p and use the description of its prismatization given at the and of my talk.
Attachment:
Akhil's proofs.pdf
Description: Adobe PDF document
Monday (May 13), 4:30 p.m, room E 206.
V.Drinfeld. Prismatization of arbitrary p-adic schemes. II.
Abstract
First, I will explain a recipe for computing the "classical
prismatization" of a p-adic scheme. Then I will describe the
prismatization of some concrete p-adic schemes.
Thursday (May 16), 4:30 p.m, room E 206.
V.Drinfeld. Prismatization of arbitrary p-adic schemes. III.
Abstract
I will discuss some of the following subjects:
de Rham and Hodge-Tate specialization of prismatic cohomology,
the prismatic cohomology of the punctured affine line.
No more meetings of the seminar this quarter.
No seminar for this quarter.
This quarter the seminar will function via Zoom.
On September 30 and in October it will be joint with Gaitsgory's seminar,
and we will meet on Wednesdays at 10 a.m. Central Time.
(I do realize that this time may be inconvenient for some of you, but it
is impossible to change it.)
Hopefully, the notes and the Zoom recording of each talk will be available
shortly after the talk at the following webpage:
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
Presumably, November will be devoted to talks by Bhatt related to
prismatic cohomology. The day and time will probably be different then.
***************************************************************
Wednesday September 30 at 10 a.m. Central Time:
David Yang (Harvard). Categorical Moy-Prasad theory.I.
(To be continued on October 7).
Here is the link for the talk:
https://harvard.zoom.us/j/95630642596?pwd=Mnk4Z0pnSC9ISHZKNFFnQjFhZ0RZdz09
Password: 041378
Abstract
We construct the categorical analogue of the depth filtration. Then we
will discuss two families of applications. The first family comes from a
new generation criterion, which we, joint with S. Raskin, apply to prove
some new results on critical level localization of modules over the affine
Lie algebra. The second family involves functional analysis of the Jacquet
functor. In particular, we analyze the failure of the categorical second
adjointness map to be an equivalence.
Wednesday September 30 at 10 a.m. Central Time:
David Yang (Harvard). Categorical Moy-Prasad theory.I.
Here is the link for the talk:
https://harvard.zoom.us/j/95630642596?pwd=Mnk4Z0pnSC9ISHZKNFFnQjFhZ0RZdz09
Password: 041378
Abstract
We construct the categorical analogue of the depth filtration. Then we
will discuss two families of applications. The first family comes from a
new generation criterion, which we, joint with S. Raskin, apply to prove
some new results on critical level localization of modules over the affine
Lie algebra. The second family involves functional analysis of the Jacquet
functor. In particular, we analyze the failure of the categorical second
adjointness map to be an equivalence.
The notes and the file of the recording of David Yang's talk are now available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ He will continue next Wednesday (Oct 7) at 10 a.m. Chicago time. Here's the Zoom link for next talk: https://harvard.zoom.us/j/94743210568?pwd=aXVWMGIxYUVnVXdyaDNJZTlMblM2UT09 Password: 941636
Wednesday (Oct 7) at 10 a.m. Chicago time: David Yang will continue his talk on Categorical Moy-Prasad theory. Zoom link: https://harvard.zoom.us/j/94743210568?pwd=aXVWMGIxYUVnVXdyaDNJZTlMblM2UT09 Password: 941636
The notes and the file of the recording of David Yang's seond talk are now
available at
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
************
Next Wednesday (Oct 14) at 10 a.m. Chicago time:
Jonathan Wang will speak about
"Spherical varieties and L-functions via geometric Langlands"
(joint work with Yiannis Sakellaridis).
Zoom link:
https://harvard.zoom.us/j/95978139120?pwd=dUMrWTZFOUp2Zmk4VkJSNXNDd1Z3dz09
Password: 740720
Abstract
The relative Langlands program, as developed by Sakellaridis and
Venkatesh, conjectures relationships between spherical varieties and
automorphic L-functions. I will give an example-based overview of these
connections, in the language of geometric Langlands when possible. In the
local setting, this is conjecturally related to the computation of nearby
cycles of an IC complex on global models of the formal arc space of a
spherical variety. I explain my joint work with Yiannis Sakellaridis where
we establish this connection and compute this nearby cycles for a nice
class of spherical varieties using the geometry of semi-infinite orbits
and affine Grassmannians.
Wednesday (Oct 14) at 10 a.m. Chicago time:
Jonathan Wang will speak about
"Spherical varieties and L-functions via geometric Langlands"
(joint work with Yiannis Sakellaridis).
Zoom link:
https://harvard.zoom.us/j/95978139120?pwd=dUMrWTZFOUp2Zmk4VkJSNXNDd1Z3dz09
Password: 740720
Abstract
The relative Langlands program, as developed by Sakellaridis and
Venkatesh, conjectures relationships between spherical varieties and
automorphic L-functions. I will give an example-based overview of these
connections, in the language of geometric Langlands when possible. In the
local setting, this is conjecturally related to the computation of nearby
cycles of an IC complex on global models of the formal arc space of a
spherical variety. I explain my joint work with Yiannis Sakellaridis where
we establish this connection and compute this nearby cycles for a nice
class of spherical varieties using the geometry of semi-infinite orbits
and affine Grassmannians.
The notes and the file of the recording of Jonathan Wang's talk are now
available at
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
************
Next Wednesday (Oct 21) at 10 a.m. Chicago time:
Xinwen Zhu (Caltech) will give his first talk on
"Coherent sheaves on the stack of Langlands parameters"
Zoom link:
https://harvard.zoom.us/j/99281000663?pwd=anVNOWN6RHg1L3U4MzRRSTRROUZCdz09
Password: 363089
Abstract
I will discuss a few recent conjectures in the arithmetic Langlands program.
In the local aspect, we discuss a categorical form of local Langlands
correspondence, which in particular predicts certain coherent sheaves on
the stack of local Langlands parameters. In the global aspect, we explain
how these coherent sheaves might be useful to understand the cohomology of
moduli of Shtukas (and Shimura varieties). I will discuss some evidences
and some examples. Some parts of the talk are based on joint work in
progress with Hemo and with Emerton.
Wednesday (Oct 21) at 10 a.m. Chicago time:
Xinwen Zhu (Caltech) will give his first talk on
"Coherent sheaves on the stack of Langlands parameters"
Zoom link:
https://harvard.zoom.us/j/99281000663?pwd=anVNOWN6RHg1L3U4MzRRSTRROUZCdz09
Password: 363089
Abstract
I will discuss a few recent conjectures in the arithmetic Langlands program.
In the local aspect, we discuss a categorical form of local Langlands
correspondence, which in particular predicts certain coherent sheaves on
the stack of local Langlands parameters. In the global aspect, we explain
how these coherent sheaves might be useful to understand the cohomology of
moduli of Shtukas (and Shimura varieties). I will discuss some evidences
and some examples. Some parts of the talk are based on joint work in
progress with Hemo and with Emerton.
The notes and the file of the recording of Xinwen Zhu's first talk on "Coherent sheaves on the stack of Langlands parameters" are now available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ He will continue next Wednesday (Oct 28) at 10 a.m. Chicago time: Zoom link: https://harvard.zoom.us/j/95839592838?pwd=NmhPdW1GQ1VVTUNGL0NkbVc3bW5zQT09 Password: 614021
Wednesday (Oct 28) at 10 a.m. Chicago time: Xinwen Zhu will give his second talk on Coherent sheaves on the stack of Langlands parameters. Zoom link: https://harvard.zoom.us/j/95839592838?pwd=NmhPdW1GQ1VVTUNGL0NkbVc3bW5zQT09 Password: 614021
The notes of Xinwen Zhu's talks (possibly in a single PDF file) will
eventually be available at the usual webpage:
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
The recording of today's talk is already available at
https://harvard.zoom.us/rec/share/QUN81akwoz6i8Qmjjs9xOaI8DvLh2vnecNvB6V_ztTvnyCKrY8GFQeiQkkXpwQIz.vSHxtzPlu_sdqv-U
________________________________________________________________________
From now on we will meet on MONDAYS at 4:10 p.m. Chicago time.
The next meeting is on November 2 (i.e., next Monday).
Bhargav Bhatt (University of Michigan) will begin his series of talks.
Title: A p-adic Riemann-Hilbert functor and vanishing theorems.
(The zoom link will be provided later).
Abstract
The goal of these talks is two-fold.
First, I'll discuss a p-adic Riemann-Hilbert functor that attaches coherent
objects to constructible F_p-sheaves on algebraic varieties over an
algebraically closed p-adic field; I'll especially focus on the good
behaviour of this functor with respect to the perverse t-structure. (This
is joint work in progress with Jacob Lurie.)
Secondly, I'll discuss a variant of the Kodaira vanishing theorem in mixed
characteristic algebraic geometry as well as some applications. This
result
relies on two ingredients: the Riemann-Hilbert functor mentioned above to
almost solve the problem, and then (log) prismatic cohomology to pass from
an almost solution to an honest solution.
The next meeting is on Monday November 2 at 4:10 p.m. Chicago time.
(NB: it seems that this Sunday the US switches to WINTER TIME. This
information is especially important for those of you who are outside of
the US.)
Nov 2 (Monday):
Bhargav Bhatt (Univ. of Michigan).
A p-adic Riemann-Hilbert functor and vanishing theorems.
Zoom link:
https://uchicago.zoom.us/j/91364347911?pwd=eTVRZWk5bmpjZFh2NllBa05VOEJGQT09
Password: 528351
Abstract
The goal of these talks is two-fold.
First, I'll discuss a p-adic Riemann-Hilbert functor that attaches coherent
objects to constructible F_p-sheaves on algebraic varieties over an
algebraically closed p-adic field; I'll especially focus on the good
behaviour of this functor with respect to the perverse t-structure. (This
is joint work in progress with Jacob Lurie.)
Secondly, I'll discuss a variant of the Kodaira vanishing theorem in mixed
characteristic algebraic geometry as well as some applications. This
result
relies on two ingredients: the Riemann-Hilbert functor mentioned above to
almost solve the problem, and then (log) prismatic cohomology to pass from
an almost solution to an honest solution.
The notes and the recording of Bhatt's yesterday talk are available at the following webpage: http://math.uchicago.edu/~amathew/GLseminarNov2020.html The same webpage will be used for his future talks. Bhatt will give his next talk Nov 9 (Monday). Zoom link: https://uchicago.zoom.us/j/98896114326?pwd=VG1DUDUyNUVoaG9NM0ZOOG5Db1NwQT09 Password: 248402
Bhargav Bhatt will give his second talk on Monday November 9. Zoom link: https://uchicago.zoom.us/j/98896114326?pwd=VG1DUDUyNUVoaG9NM0ZOOG5Db1NwQT09 Password: 248402 The notes and the recording of Bhatt's first talk are available at the following webpage: http://math.uchicago.edu/~amathew/GLseminarNov2020.html The same webpage will be used for his future talks.
Bhargav Bhatt will give his third talk on Monday November 16. Zoom link: https://uchicago.zoom.us/j/97397119978?pwd=T1FwTEJ5bVl5cmo5TFZuNzJmNkhSdz09 Password: 781188 ------------------------------------------------------------------------ The notes and the recording of Bhatt's first two talks are available at the following webpage: http://math.uchicago.edu/~amathew/GLseminarNov2020.html
The notes and the recording of all talks by Bhatt are available at the following webpage: http://math.uchicago.edu/~amathew/GLseminarNov2020.html ************** There will be no more meetings of this seminar in autumn. However, some people would possibly be interested in the following talk at Gaitsgory's seminar tomorrow (Wednesday) at 10 a.m. Chicago time. Speaker: Lin Chen (a very strong student of Gaitsgory) Title: Deligne-Lusztig duality on the category of automorphic sheaves. Zoom link: https://harvard.zoom.us/j/96179302793?pwd=TWlNRlh0Tlgyd2NMdmx0V1phZEVOUT09 Password: 759762
In January and most of February our seminar will be joint with Gaitsgory's Zoom seminar. Presumably, the first meeting will be on the week of Jan 17. If you want to attend then please participate in the Doodle poll for the time slot of the seminar. The suggested times are MWF, a 1.5 hr interval between 10am and 2pm Chicago time (18 valid options total). There are also 3 invalid options (with 3 hr interval) at the end as a result of Gaitsgory's error; please disregard them. Here's the link to the poll: https://doodle.com/poll/9n7rabix8c5us69p?utm_source=poll&utm_medium=link It would be great if you could respond by Sunday, Jan. 10. We plan 2 talks on the article "An analytic version of the Langlands correspondence for complex curves" by Etingof, Ed Frenkel, and Kazhdan, see https://arxiv.org/abs/1908.09677 Then we plan 3 talks by Dima Arinkin about his work with Gaitsgory, Kazhdan, Raskin, Rozenblyum, and Varshavsky, see https://arxiv.org/abs/2010.01906 https://arxiv.org/abs/2012.07665
In the nearest future our seminar (joint with Gaitsgory's) will meet on Wednesdays at 11:30 Chicago time. First meeting: January 27 (i.e., not very soon). The first 2 talks will be given by E. Frenkel and P. Etingof. These talks will be about their work with Kazhdan on an analytic version of the Langlands correspondence for complex curves, see https://arxiv.org/abs/1908.09677
Wednesday (Jan. 27) at 11:30 Chicago time.
Zoom link:
https://harvard.zoom.us/j/92288342938?pwd=T3VidENZV2JLanAxOCt4TUNXTUtjUT09
Password: 610510
Ed Frenkel (Berkeley) and Pavel Etingof (MIT) will give the first talk on
their work joint with Kazhdan.
(Presumably, E. Frenkel will explain the general conjecture, and Etingof
will explain what happens in the simplest nontrivial case.)
Title: An analytic version of the Langlands correspondence for complex
curves. I.
The Langlands correspondence for complex curves has been traditionally
formulated in terms of sheaves rather than functions. In 2018, Langlands
asked whether there is a function-theoretic (or analytic) version as well.
In a joint work with Kazhdan, the speakers formulated a spectral problem
for a certain commutative algebra acting on (a dense subspace of) the
Hilbert space of half-densities on Bun_G. This algebra is generated by the
global differential operators on Bun_G (the holomorphic ones, which have
been completely described by Beilinson and Drinfeld, and their complex
conjugates) and analytic Hecke operators. The authors conjecture that the
joint spectrum of this algebra can be identified with the set of opers for
the Langlands dual group of G whose monodromy is in the split real form,
up to conjugation.
Etingof plans to discuss the simplest non-trivial example of Hecke
operators over local fields, namely G=PGL(2) and genus 0 curve with 4
parabolic points. In this case the moduli space of semistable bundles
Bun_G^{ss} is P^1, and the situation is relatively well understood in the
work by Etingof, E. Frenkel and Kazhdan. He will consider the
corresponding spectral theory over C,R and non-archimedean fields and
discuss its connection with the Lame equation, Painleve VI, singular
Sturm-Liouville problems of Hilbert and Klein and previous work of
Ruijsenaars and Kontsevich.
Wednesday (Jan. 27) at 11:30 Chicago time.
Zoom link:
https://harvard.zoom.us/j/92288342938?pwd=T3VidENZV2JLanAxOCt4TUNXTUtjUT09
Password: 610510
Ed Frenkel (Berkeley) and Pavel Etingof (MIT) will give the first talk on
their work joint with Kazhdan.
(Presumably, E. Frenkel will explain the general conjecture, and Etingof
will explain what happens in the simplest nontrivial case.)
Title: An analytic version of the Langlands correspondence for complex
curves. I.
The Langlands correspondence for complex curves has been traditionally
formulated in terms of sheaves rather than functions. In 2018, Langlands
asked whether there is a function-theoretic (or analytic) version as well.
In a joint work with Kazhdan, the speakers formulated a spectral problem
for a certain commutative algebra acting on (a dense subspace of) the
Hilbert space of half-densities on Bun_G. This algebra is generated by the
global differential operators on Bun_G (the holomorphic ones, which have
been completely described by Beilinson and Drinfeld, and their complex
conjugates) and analytic Hecke operators. The authors conjecture that the
joint spectrum of this algebra can be identified with the set of opers for
the Langlands dual group of G whose monodromy is in the split real form,
up to conjugation.
Etingof plans to discuss the simplest non-trivial example of Hecke
operators over local fields, namely G=PGL(2) and genus 0 curve with 4
parabolic points. In this case the moduli space of semistable bundles
Bun_G^{ss} is P^1, and the situation is relatively well understood in the
work by Etingof, E. Frenkel and Kazhdan. In his talk Etingof will consider
the corresponding spectral theory over C,R and non-archimedean fields and
discuss its connection with the Lame equation, Painleve VI, singular
Sturm-Liouville problems of Hilbert and Klein and previous work of
Ruijsenaars and Kontsevich.
The notes and Zoom recording of today's talk are available at: http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ Next Wednesday (Feb. 3), 11:30 Chicago time: Ed Frenkel will finish his talk, and will be followed by Etingof's talk on Hecke operators over local fields (the case of G=PGL_2 and X=P^1 with 4 parabolic points). Zoom link: https://harvard.zoom.us/j/92688692019?pwd=OGROK0lHV0RPSmo2bVBrUTRNS1dSUT09 Password: 140171
Wednesday (Feb. 3), 11:30 Chicago time: Ed Frenkel will finish his talk, and will be followed by Etingof's talk on Hecke operators over local fields (the case of G=PGL_2 and X=P^1 with 4 parabolic points). Zoom link: https://harvard.zoom.us/j/92688692019?pwd=OGROK0lHV0RPSmo2bVBrUTRNS1dSUT09 Password: 140171
Edward Witten (IAS, Princeton) Title: Branes, Quantization, and Geometric Langlands https://us02web.zoom.us/j/89917028917?pwd=SVFJSWtzU2dxRWlhTldMbnZjZnRJdz09 Meeting ID: 899 1702 8917 Passcode: 698968
The slides and the Zoom recording of today's talk by E.Frenkel are
available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
On Wednesday Feb 10 at 11:30 a.m. (Chicago time)
Pavel Etingof (MIT) will continue this series of talks.
Zoom link:
https://harvard.zoom.us/j/98359695269?pwd=Q2hxckdqTlBGaXhhRDhTM0lzY2NEZz09
Password: 636827
Title: Hecke operators over local fields for G=PGL(2) on P^1 with 4
parabolic points.
Abstract: I will discuss the simplest non-trivial example of Hecke
operators over local fields, namely G=PGL(2) and genus 0 curve with 4
parabolic points. In this case the moduli space of semistable bundles
Bun_G^{ss} is P^1, and the situation is relatively well understood in our
joint work with E.Frenkel and Kazhdan. I will consider the corresponding
spectral theory over C,R and non-archimedean fields and discuss its
connection with the Lame equation, Painleve VI, singular Sturm-Liouville
problems of Hilbert and Klein and previous work of Ruijsenaars and
Kontsevich.
Wednesday (Feb 10) at 11:30 a.m. (Chicago time)
Pavel Etingof (MIT) will continue this series of talks.
This one time the seminar will slightly deviate from its usual format:
the talk will consist of 3 parts, 45 mins each, with short breaks in between.
Zoom link:
https://harvard.zoom.us/j/98359695269?pwd=Q2hxckdqTlBGaXhhRDhTM0lzY2NEZz09
Password: 636827
Title: Hecke operators over local fields for G=PGL(2) on P^1 with 4
parabolic points.
Abstract: I will discuss the simplest non-trivial example of Hecke
operators over local fields, namely G=PGL(2) and genus 0 curve with 4
parabolic points. In this case the moduli space of semistable bundles
Bun_G^{ss} is P^1, and the situation is relatively well understood in our
joint work with E.Frenkel and Kazhdan. I will consider the corresponding
spectral theory over C,R and non-archimedean fields and discuss its
connection with the Lame equation, Painleve VI, singular Sturm-Liouville
problems of Hilbert and Klein and previous work of Ruijsenaars and
Kontsevich.
Etingof's notes and the Zoom recording are now available at:
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
Furthermore, Etingof has agreed to have a Q&A session. It will
take place on Monday, February 15, 10am Chicago time. Here's the
Zoom link:
https://harvard.zoom.us/j/92484808487?pwd=eDJDMzB2U0FCN1ArWmVmc01CdkRFZz09
Password: 388379
-------------------------------------------------
Our seminar doesn't meet next week. On the other hand,
on Wednesday Feb 17 David Kazhdan is giving a talk at Minhyong Kim's
seminar at 12pm Chicago time.
Title: A proposal of a categorical construction of the algebraic version
of L^2(BunG).
Zoom link:
https://us02web.zoom.us/j/82160347061?pwd=bmo5TEo3VXdxMk1EUGdMZlFocjM1QT09
Meeting ID: 821 6034 7061
Passcode: 139770
-------------------------------------------------
Our next meeting is on Wednesday Feb 24.
Maxim Kontsevich will give a talk on "Multiplication kernels", whose
subject seems to be closely related (at least, philosophically) to the
talks by E.Frenkel and P.Etingof.
Abstract
A finite-dimensional quantum integrable system can be understood as a
structure of a commutative algebra on the "space of functions", for which
the multiplication is given by an "explicit" kernel encoded either by a
holonomic D-module on the cube of the base variety, or by a triple
correspondence with a volume element. Together with A.Odessky we found
many new explicit examples. Also, I will talk about the geometric and
operadic aspects of Sklyanin's method of separation of variables,
relevant for Hitchin systems with the gauge group GL and arbitrary
irregular singularities.
Zoom link:
https://harvard.zoom.us/j/97940324647?pwd=NWZCKzljdFRISlpxZ3E4ZC9RT3FLZz09
Password: 806651
Wednesday (Feb 24) at 11:30 a.m. Chicago time:
Maxim Kontsevich (IHES). Multiplication kernels.
Zoom link:
https://harvard.zoom.us/j/97940324647?pwd=NWZCKzljdFRISlpxZ3E4ZC9RT3FLZz09
Password: 806651
Abstract
A finite-dimensional quantum integrable system can be understood as a
structure of a commutative algebra on the "space of functions", for which
the multiplication is given by an "explicit" kernel encoded either by a
holonomic D-module on the cube of the base variety, or by a triple
correspondence with a volume element. Together with A.Odessky we found
many new explicit examples. Also, I will talk about the geometric and
operadic aspects of Sklyanin's method of separation of variables,
relevant for Hitchin systems with the gauge group GL and arbitrary
irregular singularities.
Tamas Hausel (IST Austria) Mirror symmetry for Langlands dual Higgs bundles at the tip of the nilpotent cone. Abstract: I will explain what we can prove and what we conjecture about the mirror of Hecke transformed Hitchin section motivated by symmetry ideas of Kapustin-Witten. The talk is based on arXiv:2101.08583 joint with Hitchin. Join Zoom Meeting https://upenn.zoom.us/j/91066059912?pwd=Wjg1ZnMzRWYyMzBOeUVqK1EwZVJnZz09 Meeting ID: 910 6605 9912 Passcode: 936103
The notes and the Zoom recording of Kontsevich's talk are available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ He will continue his talk on "Multiplication kernels" on Wednesday (March 3) at 11:30 a.m. Chicago time: Zoom link: https://harvard.zoom.us/j/97604492298?pwd=UjRyeHNzS0U0WGRJV3VnbkYvR0tqZz09 Password: 048333
We meet tomorrow at 11:30 Chicago time. Zoom link : https://harvard.zoom.us/j/97604492298?pwd=UjRyeHNzS0U0WGRJV3VnbkYvR0tqZz09 Password: 048333 There has been an unexpected scheduling conflict for Kontsevich, so our plan for tomorrow has changed: First, there will be an informal discussion led by Braverman and Gaistgory. As far as I understand, its goal is to explain "what is really going on" at the seminar this quarter (including Kontsevich's previous talk). Then Konstevich will join around 12:30 p.m. Chicago time. ************************** Next week (March 10) there will be a talk by Davide Gaiotto, as originally planned. Then Kontsevich will do the second half of his talk on March 17.
The scheduling conflict for Kontsevich has been cleared, so we are back to the original plan: he will start his talk *at the very beginning* of today's seminar, i.e., at 11:30 Chicago time. Zoom link : https://harvard.zoom.us/j/97604492298?pwd=UjRyeHNzS0U0WGRJV3VnbkYvR0tqZz09 Password: 048333
The notes from Kontsevich's whiteboard, the notes taken by Etingof, and the link to the Zoom recording are here: http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ Kontsevich will continue his talk on March 17 (Wednesday) at 11:30 Chicago time. Zoom link: https://harvard.zoom.us/j/92571788961?pwd=UjZ6N29EY1BHbmRZc1RvNDk4MHNWZz09 Password: 295404
The notes from Kontsevich's whiteboard, the notes taken by Etingof, and the link to the Zoom recording are here: http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ Kontsevich will continue his talk *next* Wednesday (March 10) at 11:30 Chicago time. Zoom link: https://harvard.zoom.us/j/92571788961?pwd=UjZ6N29EY1BHbmRZc1RvNDk4MHNWZz09 Password: 295404
Wednesday (March 10) at 11:30 Chicago time: Kontsevich will continue his talk. Zoom link: https://harvard.zoom.us/j/92571788961?pwd=UjZ6N29EY1BHbmRZc1RvNDk4MHNWZz09 Password: 295404 ________________________________________________________________________________________________ Attached is a preliminary draft of the paper by Kontsevich and Odesskii, which may help us to understand Kontsevich's talks.
Attachment:
Kontsevich-Odesskii-draft.pdf
Description: Adobe PDF document
The notes and recording of Kontsevich's talk are available at
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
Notes of Maxim's talks by Etingof and a preliminary version of the
Kontsevich-Odesski
paper are also available there.
On Wednesday March 17 at 11:30 Chicago time Davide Gaiotto will speak on
"Gauge theory and Vertex Algebra methods for (analytic) Geometric Langlands"
Zoom link:
https://harvard.zoom.us/j/98297283450?pwd=aUk5TFFReVAxeVFubUVKdDdjcVlDUT09
Password: 314862
Abstract
I will discuss gauge theory constructions associated to the Geometric
Langlands program. These constructions often produce vertex algebras which
can be used as concrete computational tools. The analytic version of the
Geometric Langlands program involves some novel perspectives on 2dim CFTs
which combine non-rational chiral and anti-chiral vertex algebras.
On Wednesday (March 17) at 11:30 Chicago time
Davide Gaiotto (Perimeter Institute) will speak on
"Gauge theory and Vertex Algebra methods for (analytic) Geometric Langlands"
Zoom link:
https://harvard.zoom.us/j/98297283450?pwd=aUk5TFFReVAxeVFubUVKdDdjcVlDUT09
Password: 314862
Abstract
I will discuss gauge theory constructions associated to the Geometric
Langlands program. These constructions often produce vertex algebras which
can be used as concrete computational tools. The analytic version of the
Geometric Langlands program involves some novel perspectives on 2dim CFTs
which combine non-rational chiral and anti-chiral vertex algebras.
The notes and Zoom recording of today's talk (as well as a short file written by me) have been posted at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/. Gaiotto will continue his talk on Wednesday (March 24) at 11:30 Chicago time. On the other hand, TOMORROW (Thursday) at 8 a.m. Chicago time Vincent Lafforgue will speak on a subject closely related to the talks at our seminar. More information about V.Lafforgue's talk is below: Zoom link: https://istaustria.zoom.us/j/96820239758?pwd=U20zSzJrN2xjWklIcmRmR2lVT0hhdz09 Meeting ID: 968 2023 9758 Password: 921808 Title of V.Lafforgue's talk: Classical limits for geometrizations of functoriality kernels and values of L-functions In the setting of the geometric Langlands program, it is conjectured that kernels which should give rise to Langlands functoriality, and relations between values of L-functions and some periods, exist. Some cases are known (e.g. the geometric theta correspondence and the geometrization of Rankin-Selberg integrals, due to Lysenko), the rest is mainly conjectural. However the (partly conjectural) classical limits may be described and their properties studied. In the first hour I will recall some elementary facts of symplectic geometry and the classical limit of the Langlands correspondence via the Hitchin fibration.
Here is the link to the Zoom recording of Vincent Lafforgue's yesterday talk: https://www.dropbox.com/s/k6hkwqq6yawchgk/zoom_0.mp4?dl=0 ______________________________________________ Gaiotto will continue his talk on Wednesday (March 24) at 11:30pm Chicago time. Zoom link: https://harvard.zoom.us/j/97687327058?pwd=aHkzalhxcVB4U29QQi9UNEllWmdldz09 Password: 822609
Gaiotto will continue his talk on Wednesday (March 24) at 11:30pm Chicago time. Zoom link: https://harvard.zoom.us/j/97687327058?pwd=aHkzalhxcVB4U29QQi9UNEllWmdldz09 Password: 822609
The notes and Zoom recording of Gaiotto's second talk are available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ There will be no seminar on March 31 and April 7. Presumably, we will resume on April 14 with a series of talks by Dima Arinkin.
Wednesday (April 14) at 11:30 Chicago time:
Dima Arinkin (University of Wisconsin) will give the first talk in his series
"Geometric Langlands correspondence in the restricted setting" about the
article
https://arxiv.org/abs/2010.01906
by Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum, and Varshavsky
Zoom link:
https://harvard.zoom.us/j/99594469287?pwd=NEpWQlVpV0UreGo0U2FlUWdqSXJzZz09
Password: 545805
Abstract
The geometric Langlands correspondence exists in several `flavors': both
of its sides concern topological objects (automorphic sheaves and local
systems) which, depending on the flavor, are interpreted as l-adic
sheaves, D-modules (the `de Rham flavor'), or constructible sheaves in the
classical topology ('the Betti flavor'). While the flavors are similar,
they do not match completely: for instance, the moduli space of de Rham
local systems is not isomorphic to its Betti counterpart, while the l-adic
version does not even exist (at least not as an algebraic stack).
We aim to find common ground between the different flavors, by formulating
a version of the geometric Langlands correspondence that exists in all the
settings. For instance, its de Rham and Betti versions are equivalent, its
l-adic version is no longer missing key objects/statements, and so on.
This is accomplished by modifying both sides of the conjecture: on the
automorphic side, one considers only sheaves whose singular support is
contained in the nilpotent cone, while on the Galois side, the moduli
stack of local systems is replaces by a different space: the formal moduli
stack of local systems `with restricted variation'.
I will begin by reviewing the different flavors of the Langlands
correspondence, before introducing the new `restricted' formulation. Time
permitting, I would also like to explain the spectral decomposition
theorem; among other uses, the theorem matches the `restricted variation'
of local system with a natural smoothness condition for the Hecke
functors.
Wednesday (April 14) at 11:30 Chicago time:
Dima Arinkin (University of Wisconsin) will give the first talk in his
series "Geometric Langlands correspondence in the restricted setting"
about the article
https://arxiv.org/abs/2010.01906
by Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum, and Varshavsky
Zoom link:
https://harvard.zoom.us/j/99594469287?pwd=NEpWQlVpV0UreGo0U2FlUWdqSXJzZz09
Password: 545805
Abstract
The geometric Langlands correspondence exists in several `flavors': both
of its sides concern topological objects (automorphic sheaves and local
systems) which, depending on the flavor, are interpreted as l-adic
sheaves, D-modules (the `de Rham flavor'), or constructible sheaves in the
classical topology ('the Betti flavor'). While the flavors are similar,
they do not match completely: for instance, the moduli space of de Rham
local systems is not isomorphic to its Betti counterpart, while the l-adic
version does not even exist (at least not as an algebraic stack).
We aim to find common ground between the different flavors, by formulating
a version of the geometric Langlands correspondence that exists in all the
settings. For instance, its de Rham and Betti versions are equivalent, its
l-adic version is no longer missing key objects/statements, and so on.
This is accomplished by modifying both sides of the conjecture: on the
automorphic side, one considers only sheaves whose singular support is
contained in the nilpotent cone, while on the Galois side, the moduli
stack of local systems is replaces by a different space: the formal moduli
stack of local systems `with restricted variation'.
I will begin by reviewing the different flavors of the Langlands
correspondence, before introducing the new `restricted' formulation. Time
permitting, I would also like to explain the spectral decomposition
theorem; among other uses, the theorem matches the `restricted variation'
of local system with a natural smoothness condition for the Hecke
functors.
The notes of Dima Arinkin's talk have been posted on: http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ H e will give his second talk on the Geometric Langlands correspondence in the restricted setting on Wednesday April 21 at 11:30 Chicago time Zoom link: https://harvard.zoom.us/j/96325015989?pwd=cm11RWhCTzhrTDVPZThNM3pIZWdjdz09 Password: 651401
Wednesday April 21 at 11:30 Chicago time: Dima Arinkin will give his second talk on the Geometric Langlands correspondence in the restricted setting. Zoom link: https://harvard.zoom.us/j/96325015989?pwd=cm11RWhCTzhrTDVPZThNM3pIZWdjdz09 Password: 651401
The notes and recording of today's talk by Dima Arinkin are at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ Dima will continue on Wednesday April 28 at 11:30 Chicago time. He will explain the Nadler-Yun theorem and its converse. The two theorems combined together say that an object F\in Shv(Bun_G) has nilpotent singular support if and only if the objects H_V(F)\in Shv(Bun_G \times X) are *lisse* along X. Zoom link: https://harvard.zoom.us/j/92178475094?pwd=OFUxdW82Wk0reFZwcllJZm9DUTQ1QT09 Password: 165152
Wednesday (April 28) at 11:30 Chicago time: Dima Arinkin will explain the Nadler-Yun theorem and its converse. The two theorems combined together say that an object F\in Shv(Bun_G) has nilpotent singular support if and only if the objects H_V(F)\in Shv(Bun_G \times X) are *lisse* along X. Zoom link: https://harvard.zoom.us/j/92178475094?pwd=OFUxdW82Wk0reFZwcllJZm9DUTQ1QT09 Password: 165152
The notes and recording of today's talk by Dima Arinkin are at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ Next Wednesday (May 5) at 11:30 Chicago time Alexander Braverman (University of Toronto and Perimeter Institute) will start his series of talks on his joint work with M. Finkelberg and H. Nakajima, see https://arxiv.org/abs/1706.02112 Zoom link: https://harvard.zoom.us/j/94708491059?pwd=M2dFV2k4YlNhOHJudTBiSUVZbGMwdz09 Password: 645958
On Wednesday (May 5) at 11:30 Chicago time Alexander Braverman (University of Toronto and Perimeter Institute) will start his series of talks on his joint work with M. Finkelberg and H.Nakajima, see https://arxiv.org/abs/1706.02112 Zoom link: https://harvard.zoom.us/j/94708491059?pwd=M2dFV2k4YlNhOHJudTBiSUVZbGMwdz09 Password: 645958
The recording of today's talk by Alexander Braverman is available at http://people.math.harvard.edu/~gaitsgde/GLOH_2020/ The notes will also be posted there (hopefully, tomorrow). Braverman will continue his talk next Wednesday (May 12) at 11:30 Chicago time. He will explain how ideas from [BFN] lead to the construction of quasi-classical limits of kernels for automorphic lifting. Zoom link: https://harvard.zoom.us/j/95290031254?pwd=NlY4cXd3bUI4TlVVd2NoNTZ6c1ZxZz09 Password: 754411
Wednesday (May 12) at 11:30 Chicago time: Braverman will continue his talk; he will explain how ideas from [BFN] lead to the construction of quasi-classical limits of kernels for automorphic lifting. Zoom link: https://harvard.zoom.us/j/95290031254?pwd=NlY4cXd3bUI4TlVVd2NoNTZ6c1ZxZz09 Password: 754411
The recording and notes of both talks by Braverman are available at
http://people.math.harvard.edu/~gaitsgde/GLOH_2020/
This academic year there will be no more meetings of the
Harvard-UofC geometric Langlands seminar.
(If you want to attend Gaitsgory's "office hours" seminar on Tuesdays then
you have to ask Gaitsgory to include you into his mailing list.)