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.