## Geometric Langlands Seminar

This is an archive of email messages concerning the Geometric Langlands Seminar for 2012-13.

• Date: Mon, 24 Sep 2012 13:28:47

• 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.



• Date: Sun, 30 Sep 2012 19:52:56

• 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.



• Date: Thu, 4 Oct 2012 18:24:15

• 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.



• Date: Mon, 8 Oct 2012 18:55:52

• 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.
>
>
>
>
>
>
>
>



• Date: Thu, 11 Oct 2012 18:50:18

• 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.



• Date: Wed, 17 Oct 2012 18:22:36

• 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.



• Date: Thu, 18 Oct 2012 18:51:27

• Monday (October 22), 4:30 p.m, room E 206.

Nick Rozenblyum (NWU).
Duality and D-modules via derived algebraic geometry. IV.



• Date: Mon, 22 Oct 2012 18:59:47

• 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



• Date: Mon, 29 Oct 2012 13:40:37

• 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.



• Date: Sun, 4 Nov 2012 15:33:46

• 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).



• Date: Wed, 7 Nov 2012 17:14:06

• 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).



• Date: Thu, 8 Nov 2012 18:13:16

• 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.



• Date: Mon, 12 Nov 2012 18:21:29

• 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.



• Date: Thu, 15 Nov 2012 18:30:51

• 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

• Date: Thu, 22 Nov 2012 18:29:05

• 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.



• Date: Tue, 27 Nov 2012 09:55:45

• 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.



• Date: Thu, 29 Nov 2012 18:49:57

• 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.



• Date: Wed, 5 Dec 2012 18:15:41

• 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.



• Date: Thu, 6 Dec 2012 18:52:34

• No more meetings of the Geometric Langlands seminar this quarter.



• Date: Mon, 7 Jan 2013 08:58:22

• 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.



• Date: Thu, 10 Jan 2013 20:03:48

• 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.



• Date: Tue, 15 Jan 2013 09:45:53

• 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).



• Date: Thu, 17 Jan 2013 19:25:28

• 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.



• Date: Mon, 21 Jan 2013 18:41:49

• 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.



• Date: Fri, 25 Jan 2013 11:38:40

• 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.



• Date: Fri, 25 Jan 2013 20:18:42

• 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.



• Date: Tue, 29 Jan 2013 09:39:43

• 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.



• Date: Fri, 1 Feb 2013 16:38:21

• 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.



• Date: Mon, 4 Feb 2013 18:39:18

• 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.



• Date: Thu, 7 Feb 2013 18:37:04

• 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.



• Date: Mon, 11 Feb 2013 19:16:40

• 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.



• Date: Thu, 14 Feb 2013 18:34:51

• 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 $W$.



• Date: Tue, 19 Feb 2013 18:03:52

• 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.



• Date: Thu, 21 Feb 2013 19:27:53

• 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.



• Date: Tue, 26 Feb 2013 18:36:41

• 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.



• Date: Thu, 28 Feb 2013 19:22:43

• 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.



• Date: Mon, 4 Mar 2013 20:09:13

• No more meetings of the Geometric Langlands seminar this quarter.



• Date: Mon, 1 Apr 2013 08:27:22

• 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!)



• Date: Thu, 4 Apr 2013 18:55:13

• 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.



• Date: Sun, 7 Apr 2013 12:42:54

• 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.



• Date: Mon, 8 Apr 2013 18:59:20

• 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.



• Date: Thu, 11 Apr 2013 20:01:20

• 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.



• Date: Mon, 15 Apr 2013 14:01:21

• 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.



• Date: Mon, 15 Apr 2013 20:23:38

• Thursday (April 18), 4:30 p.m, room E 206.

Ivan Losev. Introduction to categorical Kac-Moody actions.II.



• Date: Thu, 18 Apr 2013 19:19:35

• 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.



• Date: Wed, 24 Apr 2013 17:02:58

• 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.



• Date: Fri, 26 Apr 2013 18:36:03

• 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.



• Date: Mon, 29 Apr 2013 17:16:14

• 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.



• Date: Wed, 1 May 2013 18:25:18

• 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.



• Date: Thu, 2 May 2013 18:57:07

• 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.



• Date: Mon, 6 May 2013 18:00:01

• 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.



• Date: Thu, 9 May 2013 18:59:24

• 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.



• Date: Mon, 13 May 2013 19:16:37

• No more meetings of the Langlands seminar this quarter.



• Date: Mon, 23 Sep 2013 09:51:18

• 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.



• Date: Thu, 3 Oct 2013 17:06:38

• 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.



• Date: Tue, 8 Oct 2013 08:57:14

• 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.



• Date: Tue, 8 Oct 2013 19:35:33

• 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.



• Date: Thu, 10 Oct 2013 19:23:23

• 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.



• Date: Mon, 14 Oct 2013 19:33:57

• No seminar this Thursday.

Alexander Beilinson will continue on Monday (Oct 21).



• Date: Thu, 17 Oct 2013 20:21:54

• 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.



• Date: Sun, 20 Oct 2013 21:54:07

• 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.



• Date: Mon, 21 Oct 2013 19:47:55

• No seminar on Thursday.
Beilinson will continue on Monday (Oct 28).



• Date: Thu, 24 Oct 2013 17:07:19

• 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.



• Date: Mon, 28 Oct 2013 18:27:50

• 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.



• Date: Fri, 1 Nov 2013 11:43:54

• 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

• Date: Tue, 5 Nov 2013 19:23:09

• 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.



• Date: Fri, 8 Nov 2013 12:21:42

• 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

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.



• Date: Tue, 12 Nov 2013 08:12:36

• 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).



• Date: Thu, 14 Nov 2013 08:10:48

• 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

• Date: Thu, 14 Nov 2013 20:02:20

• Monday (Nov 18), 4:30 p.m, room E 206.

Danny Calegari. Fundamental groups of Kahler manifolds. II



• Date: Tue, 19 Nov 2013 10:34:20

• 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".]



• Date: Thu, 21 Nov 2013 18:24:50

• 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.



• Date: Sun, 24 Nov 2013 18:33:44

• 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.



• Date: Mon, 25 Nov 2013 20:46:52

• No more meetings of the seminar this quarter.



• Date: Thu, 2 Jan 2014 07:55:54

• 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.



• Date: Wed, 8 Jan 2014 18:03:34

• 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.



• Date: Thu, 9 Jan 2014 18:58:13

• 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.



• Date: Tue, 14 Jan 2014 08:49:13

• 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.



• Date: Thu, 16 Jan 2014 19:51:11

• 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.



• Date: Mon, 20 Jan 2014 18:57:11

• No seminar on Thursday (Jan 23).

Tamarkin will continue on Monday, January 27.



• Date: Thu, 23 Jan 2014 18:31:01

• 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.



• Date: Mon, 27 Jan 2014 19:22:59

• Monday (Feb 3), 4:30 p.m, room E 206.

Dmitry Tamarkin (NWU).  Microlocal theory of sheaves and its applications
to symplectic topology. III.



• Date: Thu, 30 Jan 2014 18:45:37

• Monday (Feb 3), 4:30 p.m, room E 206.

Dmitry Tamarkin (NWU).  Microlocal theory of sheaves and its applications
to symplectic topology. III.



• Date: Mon, 3 Feb 2014 18:39:08

• 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.



• Date: Fri, 7 Feb 2014 14:35:44

• 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.



• Date: Tue, 11 Feb 2014 10:01:31

• 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.



• Date: Fri, 14 Feb 2014 09:27:00

• 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.



• Date: Tue, 18 Feb 2014 08:41:00

• Thursday (Feb 20), 4:30 p.m, room E 206.

Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. II.



• Date: Thu, 20 Feb 2014 21:07:45

• 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.



• Date: Mon, 24 Feb 2014 17:54:47

• Thursday (Feb 27), 4:30 p.m, room E 206.

Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. III.



• Date: Thu, 27 Feb 2014 18:56:22

• 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.



• Date: Thu, 6 Mar 2014 10:51:55

• 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.



• Date: Fri, 7 Mar 2014 08:56:17

• No more seminars this quarter.

Tamarkin will explain d'Agnolo's work in spring.



• Date: Sat, 29 Mar 2014 14:17:06

• 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".



• Date: Thu, 3 Apr 2014 16:56:11

• 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.



• Date: Mon, 7 Apr 2014 18:35:51

• 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).



• Date: Mon, 14 Apr 2014 17:01:40

• 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.



• Date: Thu, 17 Apr 2014 18:47:14

• No seminar on Monday.
Xinwen Zhu will give his next talk on Thursday April 24.



• Date: Tue, 22 Apr 2014 08:53:21

• 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.



• Date: Fri, 25 Apr 2014 08:46:11

• No seminar next week.

Dima Arinkin will speak on the Monday after next week (i.e., on May 5).



• Date: Thu, 1 May 2014 17:25:35

• 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.



• Date: Tue, 6 May 2014 08:13:43

• No seminar on Thursday May 8 and Monday May 12.

Zhiwei Yun (Stanford) will speak on Thursday May 15.



• Date: Mon, 12 May 2014 20:14:55

• 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.



• Date: Thu, 15 May 2014 18:44:15

• 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.



• Date: Mon, 19 May 2014 18:43:30

• 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/



• Date: Thu, 25 Sep 2014 18:38:37

• 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).



• Date: Thu, 2 Oct 2014 17:08:37

• 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.



• Date: Mon, 6 Oct 2014 17:11:28

• 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.



• Date: Thu, 9 Oct 2014 18:44:12

• 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.



• Date: Mon, 13 Oct 2014 18:42:23

• 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

• Date: Fri, 17 Oct 2014 15:43:10

• 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).



• Date: Tue, 21 Oct 2014 09:14:12

• 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).



• Date: Thu, 23 Oct 2014 19:18:31

• 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.



• Date: Mon, 27 Oct 2014 18:48:39

• 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.



• Date: Thu, 30 Oct 2014 17:20:41

• 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.



• Date: Mon, 3 Nov 2014 18:40:31

• 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).



• Date: Thu, 6 Nov 2014 19:22:19

• 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.



• Date: Mon, 10 Nov 2014 19:42:15

• Thursday (Nov 13), 4:30 p.m, room E 206.

Sam Raskin (MIT). Chiral principal series categories. II.



• Date: Tue, 11 Nov 2014 16:26:26

• 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

• Date: Thu, 13 Nov 2014 19:29:18

• 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

• Date: Thu, 20 Nov 2014 16:41:46

• 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.



• Date: Mon, 24 Nov 2014 19:20:44

• Monday (Dec 1), 4:30 p.m, room E 206.

Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical
models. II.



• Date: Thu, 27 Nov 2014 11:23:45

• Monday (Dec 1), 4:30 p.m, room E 206.

Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical
models. II.



• Date: Mon, 1 Dec 2014 18:51:44

• No more meetings of the seminar this quarter.



• Date: Sat, 3 Jan 2015 10:45:13

• 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).



• Date: Tue, 6 Jan 2015 19:56:36

• 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.



• Date: Thu, 8 Jan 2015 19:35:18

• 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).



• Date: Mon, 12 Jan 2015 19:33:49

• 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.



• Date: Tue, 20 Jan 2015 08:50:22

• 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.



• Date: Fri, 23 Jan 2015 09:50:59

• 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.



• Date: Mon, 26 Jan 2015 20:18:09

• 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.



• Date: Thu, 29 Jan 2015 19:10:23

• 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.



• Date: Mon, 2 Feb 2015 17:26:43

• 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.



• Date: Fri, 6 Feb 2015 18:49:43

• 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>



• Date: Mon, 9 Feb 2015 18:58:31

• Ivan Mirkovic will continue his talk on Zastava spaces on
Thursday (Feb 12), 4:30 p.m, room E 206.



• Date: Thu, 12 Feb 2015 19:36:10

• Ivan Mirkovic will continue his talk on Zastava spaces on
Monday (Feb 16), 4:30 p.m, room E 206.



• Date: Tue, 17 Feb 2015 09:07:47

• 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.



• Date: Thu, 19 Feb 2015 17:18:53

• 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.



• Date: Mon, 23 Feb 2015 18:32:50

• 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.



• Date: Thu, 26 Feb 2015 21:44:07

• 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.

`