## 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.



• Date: Tue, 3 Mar 2015 10:22:26

• No more meetings of the seminar this quarter.

**********
Jochen Heinloth from University of Essen will give a series of lectures
starting on this Friday, March 6, 2-3PM, room E203. The topic of his
lectures will be:

An introduction to the P=W conjecture and related conjectures of Hausel.

Abstract

I will try to explain the work of Hausel and Rodriguez-Villegas and de
Cataldo-Hausel-Migliorini resulting in a series of conjectures
on the global geometry of moduli spaces of Higgs bundles.

The starting point will be the different algebraic structures on the
manifold underlying the moduli space of Higgs bundles on a curve. Hausel
and Rodriguez-Villegas managed to do point counting arguments in one of
the complex structures (the character variety) and using this, they found
interesting properties of the cohomology that are reminiscent of
properties of intersection cohomology. This finally led de
Cataldo-Hausel-Migliorini to propose the P=W conjecture which they could
prove in some cases.



• Date: Fri, 27 Mar 2015 17:17:31

• No meetings of the seminar during the first week of the quarter.
First meeting: Monday April 6 (4:30 p.m, room E 206).

On April 6 Nick Rozenblyum will give the first talk in a series devoted to
his joint work with David Ayala and John Francis.

Title: Higher categories and manifold topology. I.

Abstract

Over the past few decades, there has been a fruitful interplay between
manifold topology and (higher) category theory. I will give an overview of
some of these connections, and discuss joint work with David Ayala and
John Francis, which describes higher categories in terms of the topology
of stratified manifolds. This approach provides a precise dictionary
between manifold topology and higher category theory, and makes numerous
connections between the two manifest.



• Date: Fri, 3 Apr 2015 14:15:31

• Monday (April 6), 4:30 p.m, room E 206.

Nick Rozenblyum. Higher categories and manifold topology.

This is the first talk in a series. Most probably, Nick will also speak on
Thursday April 9. The next talks will be given by David Ayala, see
http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands

Abstract

Over the past few decades, there has been a fruitful interplay between
manifold topology and (higher) category theory. I will give an overview of
some of these connections, and discuss joint work with David Ayala and
John Francis, which describes higher categories in terms of the topology
of stratified manifolds. This approach provides a precise dictionary
between manifold topology and higher category theory, and makes numerous
connections between the two manifest.



• Date: Mon, 6 Apr 2015 19:08:32

• Thursday (Apr 9), 4:30 p.m, room E 206.

Nick Rozenblyum. Higher categories and manifold topology.II.



• Date: Thu, 9 Apr 2015 19:37:33

• No seminar on Monday (Apr 13).

On Thursday Apr 16 David Ayala will give his first talk.
Title: Higher categories are sheaves on manifolds.

Abstract

This project is an effort to merge higher algebra/category theory and
differential topology. As an outcome, information flows in both
directions: coherent constructions of manifold and embedding invariants
from higher algebraic/categorical data, such as that of a representation
of a quantum group lending to knot invariants; deformations of higher
algebraic/categorical parameters indexed by manifolds, such as Hochschild
(co)homology.

The talks will be framed by one main result, and a couple formal
applications thereof.   The main construction is factorization homology
with coefficients in higher categories.  The body of the talks will focus
on essential aspects of our definitions that facilitate the coherent
cancelations that support our main result.

This is joint work with John Francis and Nick Rozenblyum.



• Date: Mon, 13 Apr 2015 16:46:12

• Thursday (Apr 16), 4:30 p.m, room E 206.

David Ayala (Montana State University).
Higher categories are sheaves on manifolds. I.

Abstract

This project is an effort to merge higher algebra/category theory and
differential topology. As an outcome, information flows in both
directions: coherent constructions of manifold and embedding invariants
from higher algebraic/categorical data, such as that of a representation
of a quantum group lending to knot invariants; deformations of higher
algebraic/categorical parameters indexed by manifolds, such as Hochschild
(co)homology.

The talks will be framed by one main result, and a couple formal
applications thereof.   The main construction is factorization homology
with coefficients in higher categories.  The body of the talks will focus
on essential aspects of our definitions that facilitate the coherent
cancelations that support our main result.

This is joint work with John Francis and Nick Rozenblyum.



• Date: Thu, 16 Apr 2015 18:46:30

• Monday (Apr 20), 4:30 p.m, room E 206.

David Ayala. Higher categories are sheaves on manifolds. II.



• Date: Mon, 20 Apr 2015 18:56:41

• Thursday (April 23), 4:30 p.m, room E 206.

Bhargav Bhatt (University of Michigan). Integral p-adic Hodge theory.

Abstract

Let X be a smooth and proper scheme over the ring of integers in a p-adic
field. Classical p-adic Hodge theory relates the etale and de Rham
cohomologies of X: the theories are naturally identified after extending
scalars to a suitable ring of periods constructed by Fontaine. This
isomorphism is compatible with Galois actions, and thus plays a crucial
role in our understanding of Galois representations. This identification,
however, neglects all torsion phenomena as the period ring is a Q-algebra.

In my talk, I will briefly recall this rational story, and then describe a
new "comparison" between these two cohomology theories that works
integrally: we will realize the de Rham cohomology of X as a
specialization of the etale cohomology, integrally, over a 2-parameter
base. As an application, we deduce the optimal result relating torsion in
the two theories: the torsion in de Rham cohomology is an upper bound for
the torsion in etale cohomology (and the inequality can be strict). This
inequality can be used to explain some of the "pathologies" in de Rham
cohomology in characteristic p.

(Based on joint work in progress with Morrow and Scholze.)



• Date: Fri, 24 Apr 2015 13:12:37

• No seminar next week.



• Date: Sun, 3 May 2015 09:57:23

• 1. No seminar this week.
2. Joseph Bernstein will be visiting our department starting from
Wednesday May 6. He will give a seminar talk on Monday May 11 and at least
one talk after that (on May 18 and maybe May 21).

Yiannis Sakellaridis will be visiting us on May 13-16; he will speak on
May 14.

The titles and abstracts can be found at
http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands

3. A mysterious theorem is formulated on p.2 of the article
http://arxiv.org/pdf/math/0701615v3.pdf
by Kumar, Lusztig, and Dipendra Prasad. As explained there, the theorem is
proved in Jantzen's Ph.D. thesis (1973). An equivalent formulation is
given in the Corollary (also on p.2). DOES ANYBODY KNOW A CONCEPTUAL
EXPLANATION of the result? (E.g., is there any categorical statement
behind it?)

The result is as follows. Let G be a connected simply connected
almost-simple group equipped with a pinning (English translation of
Bourbaki's "epinglage"). Let \sigma be a nontrivial automorphism of the
Dynkin diagram, then \sigma acts on G. Let G^\sigma denote the subgroup of
fixed points. Let G_\sigma denote the simply connected group whose root
system is dual to that of G^\sigma . The Corollary on p.2 says that the
\sigma-characters of G are equal to the characters of G_\sigma . (This is
mysterious because there is no apparent relation between G and G_\sigma
and also because passing to the dual root system is just a formal
combinatorial operation).



• Date: Sat, 9 May 2015 12:31:34

• Monday (May 11), 4:30 p.m, room E 206.

Joseph Bernstein (Tel Aviv University). Stacks in Representation Theory.
(What is a representation of an algebraic group?)

Abstract

I will discuss a new approach to representation theory of algebraic groups.

In the usual approach one starts with an algebraic group G over some local
(or finite) field F, considers the group G(F) of its F-points as a
topological group and studies some category Rep (G(F)) of continuous
representations of the group G(F).

I will argue that more correct objects to study are some kind of sheaves
on the stack BG corresponding to the group G.

I will show that this point of view naturally requires to change the
formulation of some basic problems in Representation Theory. In
particular, this approach might explain the appearance of representations
of all pure forms of a group G in Vogan's formulation of Langlands'
correspondence.



• Date: Mon, 11 May 2015 18:37:21

• Thursday (May 14), 4:30 p.m, room E 206.

Yiannis Sakellaridis (Rutgers University). Spectral decomposition on
homogeneous spaces.

Abstract

I will present results from my joint work with Venkatesh, Delorme and
Harinck on harmonic analysis on homogeneous spaces. These results have
been established for spherical homogeneous spaces over p-adic fields, but
most of their analogs exist for automorphic functions, and the talk will
attempt to cover those as well (especially in the function field case,
where technicalities due to Archimedean places do not arise).

The general structure of these results is the following: For a given
G-space X, there are "simpler" G-spaces X_i (of the same dimension but
with more symmetries, i.e. non-trivial groups of G-automorphisms) such
that functions on X decompose into a "discrete modulo automorphisms" part
plus a homomorphic image of the "discrete modulo automorphisms" part of
the spaces X_i. There are smooth and L^2 versions of this story, and for
the former the word "discrete" should be replaced by "cuspidal".

The talk will emphasize general principles (largely based on ideas of
Joseph Bernstein) that give rise to the same kind of decomposition
irrespective of the space, as well as points in the method that have still
not been clarified enough.



• Date: Wed, 13 May 2015 14:57:15

• SPECIAL SEMINAR
on Tuesday (May 19), 1:30 p.m, room E 206.

Pham H. Tiep (University of Arizona).
Bounding character values of finite groups of Lie type.

Abstract

Let G be a finite group of Lie type. In spite of many foundational results
on complex representation theory of G, several questions on character
values still remain open. One such question, essential for various
applications, including random walks and word maps on finite simple
groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and
for all irreducible complex characters \chi of G. In the case of symmetric
groups, this problem has been solved by Larsen and Shalev. We will discuss
recent progress on this problem for finite groups of Lie type, obtained in
joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.



• Date: Thu, 14 May 2015 19:01:02

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

Joseph Bernstein (Tel Aviv University).
Convexity and subconvexity bounds for Automorphic Periods
(joint work with A. Reznikov).

Abstract

In my lecture I will discuss basic problems related to bounds on
periods of automorphic functions.  One of my goals is to discuss
the  insight into these bounds given by relation of periods to special
values of L-functions (for example this predicts some convexity and
subconvexity bounds on periods coming from L-function theory).
I describe a method based on Representation Theory of real groups
that allows to analyze such bounds.

I will concentrate on two very concrete problems.
Let Y be a compact Riemannian surface of constant
curvature -1. A Maass form is a function f on Y that
is an eigenfunction of the Laplace operator D.

Problem 1 (Fourier expansion). Fix a closed geodesic C
in Y , restrict some Maass form f to the circle C and
consider Fourier coefficients a_k of this function.
How to give bounds on these coefficients when k tends
to infinity ?

Problem 2 (Triple product). Let p be a product of two
Maass forms.
How to give bound on the scalar product <p, f> of the
function p  with a  Maass form f in terms of the
eigenvalue of f when this eigenvalue tends to infinity ?



• Date: Mon, 18 May 2015 12:05:58

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

Joseph Bernstein (Tel Aviv University).
Convexity and subconvexity bounds for Automorphic Periods
(joint work with A. Reznikov).

Abstract

In my lecture I will discuss basic problems related to bounds on
periods of automorphic functions.  One of my goals is to discuss
the  insight into these bounds given by relation of periods to special
values of L-functions (for example this predicts some convexity and
subconvexity bounds on periods coming from L-function theory).
I describe a method based on Representation Theory of real groups
that allows to analyze such bounds.

I will concentrate on two very concrete problems.
Let Y be a compact Riemannian surface of constant
curvature -1. A Maass form is a function f on Y that
is an eigenfunction of the Laplace operator D.

Problem 1 (Fourier expansion). Fix a closed geodesic C
in Y , restrict some Maass form f to the circle C and
consider Fourier coefficients a_k of this function.
How to give bounds on these coefficients when k tends
to infinity ?

Problem 2 (Triple product). Let p be a product of two
Maass forms.
How to give bound on the scalar product <p, f> of the
function p  with a  Maass form f in terms of the
eigenvalue of f when this eigenvalue tends to infinity ?



• Date: Mon, 18 May 2015 18:53:55

• SPECIAL SEMINAR
on Tuesday (May 19), 1:30 p.m, room E 206.

Pham H. Tiep (University of Arizona).
Bounding character values of finite groups of Lie type.

Abstract

Let G be a finite group of Lie type. In spite of many foundational results
on complex representation theory of G, several questions on character
values still remain open. One such question, essential for various
applications, including random walks and word maps on finite simple
groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and
for all irreducible complex characters \chi of G. In the case of symmetric
groups, this problem has been solved by Larsen and Shalev. We will discuss
recent progress on this problem for finite groups of Lie type, obtained in
joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.



• Date: Tue, 19 May 2015 16:11:33

• Thursday (May 21), 4:30 p.m, room E 206.

Joseph Bernstein (Tel Aviv University). Convexity and subconvexity bounds
of Automorphic Periods. II.



• Date: Thu, 21 May 2015 20:12:01

• No more meetings of the seminar in this quarter.



• Date: Thu, 24 Sep 2015 16:57:51

• As usual, the seminar will meet on Mondays and/or Thursdays in room E206
at 4:30 p.m.

We begin on Thursday Oct 1 with Peter Sarnak’s talk on the very important
notion of analytic conductor in the theory of automorphic forms, see
http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands
for more details.

On Thursday Oct 8 Laurent Fargues will begin his mini-course
“Geometrization of the local Langlands correspondence”, in which he will
explain his new and very exciting conjectures.
(As far as I understand, they are in the spirit of the global unramified
geometric Langalnds conjecture, but instead of usual curves he considers
the “curve” that he and Fontaine had associated to an arbitrary
non-Archimedean local field E. Because of that, he ends up with
conjectures that would imply the local Langlands conjecture for E.)

Presumably, Fargues will give 8 lectures. The title and abstract of his
course can be found at
http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands



• Date: Mon, 28 Sep 2015 16:34:04

• Thursday (Oct 1), 4:30 p.m, room E 206.

Peter Sarnak (IAS). The analytic conductor in the theory of automorphic
forms.

Abstract

The analytic conductor of an automorphic cusp form on GL(n)  over a number
field is a measure of its complexity: especially in connection with the
corresponding L-function. We review some of the definitions, properties
and central role played by the conductor. If time permits we discuss some
recent applications to fast computations of epsilon factors and the Mobius
function.



• Date: Fri, 2 Oct 2015 11:13:37

• No seminar on Monday (Oct.5).

On Thursday (Oct.8) Laurent Fargues will begin his mini-course on
Geometrization of the local Langlands correspondence



• Date: Mon, 5 Oct 2015 16:55:18

• Thursday (October 8), 4:30 p.m, room E 206.

Laurent Fargues (Institut de Mathematiques de Jussieu) will begin his
mini-course on
Geometrization of the local Langlands correspondence

I will explain a recent conjecture giving a geometrization of the local
Langlands correspondence over a non-archimedean local field. The purpose
is to explain the precise statement of the conjecture and evidences for
it. For this I will introduce the objects that appear, in particular the
curve defined and studied in our joint work with Fontaine, the structure
of G-bundles on this curve and the basic properties of the associated
stack of G-bundles.

Presumably, there will be 8 lectures in the mini-course.



• Date: Fri, 9 Oct 2015 08:59:52

• No seminar on Monday Oct 12.

Laurent Fargues will give his second lecture next Thursday (Oct 15). I
will inform you when the notes of his first lecture become available.

Program for the rest of October:
Laurent Fargues will speak on Oct 15, 22, 26.
Bhargav Bhatt will give a talk on Oct 19.

Information on the Oberwolfach workshop mentioned by Fargues is at
www.mfo.de/occasion/1614/www_view



• Date: Fri, 9 Oct 2015 15:40:07

• Laurent Fargues says that to understand his yesterday lecture, one can
read one of the following articles available at his homepage
http://webusers.imj-prg.fr/~laurent.fargues/Publications.html

1. Factorization of analytic functions in mixed characteristic
2. Vector bundles and p-adic Galois representations
3. Vector bundles on curves and p-adic Hodge theory

The first article is the most elementary, and Fargues says that already
that article covers the material of his yesterday lecture.

(The third article is the most advanced.)



• Date: Sun, 11 Oct 2015 09:10:03

• The notes of the first lecture by Fargues are at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf
The notes were made by Sean Howe. He says:

> As the lectures continue I'll update that file. Already in the first
> lecture there are probably some typos I've missed; I'll correct them as I
> or others find them.



• Date: Mon, 12 Oct 2015 18:45:11

• Laurent Fargues will give his second lecture on
Thursday (Oct 15), 4:30 p.m, room E 206.



• Date: Thu, 15 Oct 2015 18:47:11

• Monday (Oct 19), 4:30 p.m, room E 206.

Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian.

Abstract
The affine Grassmannian is an ind-variety over a field k that parametrizes
k[[t]]-lattices in a fixed k((t))-vector space. I will explain the
construction of a p-adic analog, i.e., an ind-scheme over F_p that
parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on
recent work of X. Zhu. The construction takes place in the world of
algebraic geometry with perfect schemes, and a large portion of the talk
will be devoted to explaining certain surprisingly nice features of this
world. (This talk is based on joint work with Peter Scholze.)



• Date: Thu, 15 Oct 2015 20:29:19

• Monday (Oct 19), 4:30 p.m, room E 206.

Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian.

Abstract
The affine Grassmannian is an ind-variety over a field k that parametrizes
k[[t]]-lattices in a fixed k((t))-vector space. I will explain the
construction of a p-adic analog, i.e., an ind-scheme over F_p that
parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on
recent work of X. Zhu. The construction takes place in the world of
algebraic geometry with perfect schemes, and a large portion of the talk
will be devoted to explaining certain surprisingly nice features of this
world. (This talk is based on joint work with Peter Scholze.)



• Date: Mon, 19 Oct 2015 18:57:19

• Laurent Fargues will give his third lecture on
Thursday (Oct 22), 4:30 p.m, room E 206.



• Date: Tue, 20 Oct 2015 08:33:58

• The notes at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf
now include the second lecture by Fargues.

Fargues will give his third lecture on Thursday (usual time and place).



• Date: Thu, 22 Oct 2015 18:36:21

• Laurent Fargues will give his next lecture on
MONDAY (Oct 26), 4:30 p.m, room E 206.



• Date: Mon, 26 Oct 2015 18:58:47

• No seminar on Thursday.

The notes from Fargues' Thursday lecture are now online at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf

Laurent Fargues will give his next lecture on
Monday (Nov 2), 4:30 p.m, room E 206.



• Date: Sat, 31 Oct 2015 12:15:50

• Laurent Fargues will give his next lecture on
Monday (Nov 2), 4:30 p.m, room E 206.

*******
Many of us know the name of Andrei Zelevinsky, who worked in
representation theory (including the Langlands program) and combinatorics.
He died a few years ago, when he was only 60.

I just learned that Northeastern University has established a prestigious
post-doctoral position named in Zelevinsky's memory. More information (in
particular, instructions how to donate) can be found at
http://avzel.blogspot.com/2015/10/andrei-zelevinsky-research-instructor.html

http://zelevinsky.com/Zelevinsky_Fund_Letter.pdf

Andrei was a very good mathematician and a very good man.

(We first met at a mathematical olympiad when he was 16 and I was 15.
At that time he was very impressed by Vilenkin's book on combinatorics.
Later combinatorics became a major area of his research...)



• Date: Sun, 1 Nov 2015 19:13:00

• The notes of Fargues' Monday lecture are now online at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf

Laurent Fargues will give his next lecture
tomorrow, i.e., Monday (Nov 2), 4:30 p.m, room E 206.



• Date: Mon, 2 Nov 2015 08:25:27

• Laurent Fargues has to CANCEL his lecture today.



• Date: Tue, 3 Nov 2015 10:49:46

• Laurent Fargues will give his next lecture
on Thursday (Nov 5), 4:30 p.m, room E 206.



• Date: Thu, 5 Nov 2015 20:01:16

• Laurent Fargues will give his next lecture
on Monday (Nov 9), 4:30 p.m, room E 206.



• Date: Mon, 9 Nov 2015 18:53:17

• Laurent Fargues will give his next lecture
on Thursday (Nov 12), 4:30 p.m, room E 206.



• Date: Thu, 12 Nov 2015 18:38:19

• Monday (Nov 16) and Thursday (Nov 19), 4:30 p.m, room E 206.

Akshay Venkatesh (Stanford University). Motivic cohomology and the
cohomology of arithmetic groups.

Abstract

The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke
action. The same system of Hecke eigenvalues will usually occur in
multiple  cohomological degrees. This suggests the existence of extra
endomorphisms of the cohomology that commute with the Hecke operators but
shift cohomological degree. (In the Shimura case these are provided by
"Lefschetz operators" but the situation in general is much more
interesting.)

I will explain a conjecture that, in fact, the motivic cohomology of the
associated motives act on the cohomology and provide these extra
endomorphisms. (According to the Langlands program, a Hecke eigenclass
occurring in cohomology should give a system of motives, indexed by
representations of the dual group. The "associated" motives we need are
the ones associated to the adjoint representation of the dual group.)

This structure should exist at the level of cohomology with
Q-coefficients, but I don't know how to construct it.
However,  one can construct the corresponding action on cohomology with
real or p-adic coefficients, using the corresponding regulator map on the
motivic cohomology, and then try to get evidence that it preserves
Q-structures.

LECTURE ONE:  I will explain the overall conjecture and how to construct
the action with real coefficients.  This is joint work with Prasanna. In
particular, we  are able to give evidence that the action preserves
Q-structures, basically by relating it to the theory of periods of
automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture),
and also by using analytic torsion.

LECTURE TWO:  I will explain the story with p-adic coefficients.  The
story here is algebraically richer;  there are two related way of
constructing extra endomorphisms of the cohomology. One is via a derived
version of the Hecke algebra and one is via a derived version of Mazur's
Galois deformation ring (joint with Soren Galatius).    Here there is no
evidence, at present, that this preserves Q-structures.



• Date: Wed, 18 Nov 2015 08:16:13

• Thursday (Nov 19), 4:30 p.m, room E 206.

Akshay Venkatesh (Stanford University). Motivic cohomology and the
cohomology of arithmetic groups. II.

Abstract

The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke
action. The same system of Hecke eigenvalues will usually occur in
multiple  cohomological degrees. This suggests the existence of extra
endomorphisms of the cohomology that commute with the Hecke operators but
shift cohomological degree. (In the Shimura case these are provided by
"Lefschetz operators" but the situation in general is much more
interesting.)

I will explain a conjecture that, in fact, the motivic cohomology of the
associated motives act on the cohomology and provide these extra
endomorphisms. (According to the Langlands program, a Hecke eigenclass
occurring in cohomology should give a system of motives, indexed by
representations of the dual group. The "associated" motives we need are
the ones associated to the adjoint representation of the dual group.)

This structure should exist at the level of cohomology with
Q-coefficients, but I don't know how to construct it.
However,  one can construct the corresponding action on cohomology with
real or p-adic coefficients, using the corresponding regulator map on the
motivic cohomology, and then try to get evidence that it preserves
Q-structures.

LECTURE ONE:  I will explain the overall conjecture and how to construct
the action with real coefficients.  This is joint work with Prasanna. In
particular, we  are able to give evidence that the action preserves
Q-structures, basically by relating it to the theory of periods of
automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture),
and also by using analytic torsion.

LECTURE TWO:  I will explain the story with p-adic coefficients.  The
story here is algebraically richer;  there are two related way of
constructing extra endomorphisms of the cohomology. One is via a derived
version of the Hecke algebra and one is via a derived version of Mazur's
Galois deformation ring (joint with Soren Galatius).    Here there is no
evidence, at present, that this preserves Q-structures.



• Date: Thu, 19 Nov 2015 18:33:11

• Laurent Fargues will speak
on Monday (Nov 23), 4:30 p.m, room E 206.

Notes from the 5th and 6th lectures by Fargues are now available at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf
Notes from the 7th lecture are expected to appear relatively soon (maybe
on Sunday).



• Date: Mon, 23 Nov 2015 08:48:52

• The notes of Fargues' November 12 lecture are now online at
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf

He will speak today at the usual time and place.



• Date: Mon, 23 Nov 2015 18:56:27

• Laurent Fargues will continue
tomorrow (i.e.,  Tuesday Nov 24), 1:30 p.m, room E 202.



• Date: Tue, 24 Nov 2015 15:07:48

• No more meetings of the seminar this quarter.

Happy Thanksgiving!



• Date: Sat, 2 Jan 2016 12:38:18

• The first meeting of the seminar is on
Thursday (Jan 7), 4:30 p.m, room E 206.

Alexander Beilinson will begin his series of talks on
The singular support and characteristic cycle of etale sheaves.

Abstract

Singular support and characteristic cycle are fundamental notions of the
theory of D-modules; they were rendered into the setting of constructible
sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira
in their book on microlocal theory of sheaves. This series of talks treats
the case of etale sheaves on varieties over a field of finite
characteristic studied recently by
T.Saito, http://lanl.arxiv.org/abs/1510.03018
and myself, http://lanl.arxiv.org/abs/1505.06768.
In dimension one the story amounts to the classical
Grothendieck-Ogg-Shafarevich formula.

In the first talk I will remind, as a warm-up, the classical D-module and
Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar
old observation by Deligne about non-integrability of characteristics, and
introduce the main tool of Brylinski's Radon transform. If time permits, I
will give a proof of the basic upper estimate on the dimension of the
singular support.



• Date: Wed, 6 Jan 2016 11:16:13

• Thursday (Jan 7), 4:30 p.m, room E 206.

Alexander Beilinson will begin his series of talks on
The singular support and characteristic cycle of etale sheaves.

Abstract

Singular support and characteristic cycle are fundamental notions of the
theory of D-modules; they were rendered into the setting of constructible
sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira
in their book on microlocal theory of sheaves. This series of talks treats
the case of etale sheaves on varieties over a field of finite
characteristic studied recently by
T.Saito, http://lanl.arxiv.org/abs/1510.03018
and myself, http://lanl.arxiv.org/abs/1505.06768.
In dimension one the story amounts to the classical
Grothendieck-Ogg-Shafarevich formula.

In the first talk I will remind, as a warm-up, the classical D-module and
Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar
old observation by Deligne about non-integrability of characteristics, and
introduce the main tool of Brylinski's Radon transform. If time permits, I
will give a proof of the basic upper estimate on the dimension of the
singular support.



• Date: Thu, 7 Jan 2016 20:32:06

• Deligne's letter mentioned by Sasha is at
http://math.uchicago.edu/~drinfeld/Deligne's_letter_SingSupp.pdf

******
Next talk:
Monday (Jan 11), 4:30 p.m, room E 206.

Daniil Rudenko (Moscow). Goncharov conjectures and functional equations
for polylogarithms.

Abstract

Classical polylogarithms and functional equations which these functions
satisfy have been studied since the beginning of the 19th century.
Nevertheless, the structure of these equations is still understood very
poorly. I will explain an approach to this subject, based on the link
between polylogarithms and mixed Tate motives.

A substantial part of the talk will be devoted to the explanation of this
link, provided by Goncharov Conjectures. After that, I will present some
results about functional equations which can be proved unconditionally. If
time permits, I will finish with another application of this circle of
ideas to scissor congruence theory.



• Date: Sun, 10 Jan 2016 10:40:33

• A few years ago I wrote some notes for myself, which can be found here:
http://www.math.uchicago.edu/~drinfeld/Cotangent_notes-2011/Notes-2011.pdf

They are closely related to Sasha's Thursday talk. A brief explanation of
the subject of my notes and the relation with Sasha's talk can be found
here:

******
As already announced, on Monday (Jan 11) Daniil Rudenko will speak on
Goncharov conjectures and functional equations for polylogarithms.



• Date: Mon, 11 Jan 2016 18:38:20

• Thursday (Jan 7), 4:30 p.m, room E 206.

Alexander Beilinson will give his second talk on
The singular support and characteristic cycle of etale sheaves.



• Date: Tue, 12 Jan 2016 08:18:58

• Thursday (Jan 14), 4:30 p.m, room E 206.

Alexander Beilinson will give his second talk on
The singular support and characteristic cycle of etale sheaves.



• Date: Thu, 14 Jan 2016 19:17:12

• No seminar on Monday.

Beilinson will continue on Thursday (Jan 21).



• Date: Thu, 21 Jan 2016 18:43:17

• Beilinson will continue on Monday (Jan 25).



• Date: Sat, 23 Jan 2016 14:47:41

• Sasha will speak on Monday (as announced before).

Here is Sean Howe's message about his notes of Fargues' lectures.
--------------------------------------------------------------------------
Subject: Full notes available
From:    "Sean Howe" <seanpkh@gmail.com>
Date:    Sat, January 23, 2016 1:12 pm
To:      "Vladimir G. Drinfeld" <drinfeld@math.uchicago.edu>
--------------------------------------------------------------------------

Hi all,

The full notes from Fargues' lectures last quarter are now available on my
website:
http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf

I am sorry for the very long delay in posting the final two lectures; I had
some unresolved questions about the last lecture and had trouble finding
the time to understand them over winter break.

The notes are a preliminary version still -- eventually I will add more
references, clean up typos, and most importantly fix some remaining issues
with the final section on local-global compatibility (which is still very
rough -- I've put a warning about this in the notes). But I thought it'd be
better to get them up now and then worry about that in the future! Please
let me know of any changes that should be made and I will try to
incorporate them in a more timely fashion.

Thanks!

Best,
Sean



• Date: Mon, 25 Jan 2016 19:10:36

• Beilinson will continue on Thursday (Jan 28).



• Date: Thu, 28 Jan 2016 18:40:18

• No seminar on Monday. The next meeting is on Thursday.
The title and abstract of the talk are below.
("Schober" is a German word; one of its meanings is "haystack". My guess
is that the Schobers from Kapranov's talk are "kind of stacks".)

Thursday (Feb 4), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).  Perverse Schobers.

Abstract

I will explain the project of developing a theory of "perverse sheaves of
triangulated categories". One motivation for it is the desire of
introducing coefficients in the definition of Fukaya categories (which are
categorical analogs of (co)homology with constant coefficients). Since
perverse sheaves are complexes of sheaves and not just sheaves, their
categorical analogs are not obvious. Nevertheless in several interesting
cases the definition can be made, and we can make the first steps in the
cohomological formalism. The talk is based on joint work with T.
Dyckerhoff, V. Schechtman and Y. Soibelman.



• Date: Mon, 1 Feb 2016 20:16:05

• Thursday (Feb 4), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).  Perverse Schobers.

Abstract

I will explain the project of developing a theory of "perverse sheaves of
triangulated categories". One motivation for it is the desire of
introducing coefficients in the definition of Fukaya categories (which are
categorical analogs of (co)homology with constant coefficients). Since
perverse sheaves are complexes of sheaves and not just sheaves, their
categorical analogs are not obvious. Nevertheless in several interesting
cases the definition can be made, and we can make the first steps in the
cohomological formalism. The talk is based on joint work with
T.Dyckerhoff, V.Schechtman and Y.Soibelman.



• Date: Fri, 5 Feb 2016 19:21:18

• No seminar on Monday.

Thursday (Feb 11), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).
Homotopy Lie algebras associated to secondary polytopes.

Abstract

Motivated by the work of Gaiotto-Moore-Witten on the "algebra of
infrared", we construct a homotopy Lie algebra associated to the secondary
polytope of a finite set  A of points in the  n-dimensional Euclidean
space.  While the construction can be made for any n (and leads to
E_n-algebras), the case of "physical" interest is when A consists of
critical values of a holomorphic Morse function. The talk is based on
joint work with M. Kontsevich and Y. Soibelman.



• Date: Mon, 8 Feb 2016 18:11:00

• Thursday (Feb 11), 4:30 p.m, room E 206.
Mikhail Kapranov (Kavli Institute, Japan).
Homotopy Lie algebras associated to secondary polytopes.

Abstract

Motivated by the work of Gaiotto-Moore-Witten on the "algebra of
infrared", we construct a homotopy Lie algebra associated to the secondary
polytope of a finite set  A of points in the  n-dimensional Euclidean
space.  While the construction can be made for any n (and leads to
E_n-algebras), the case of "physical" interest is when A consists of
critical values of a holomorphic Morse function. The talk is based on
joint work with M. Kontsevich and Y. Soibelman.



• Date: Thu, 11 Feb 2016 18:55:18

• Monday (Feb 15), 4:30 p.m, room E 206.
Kiran Kedlaya (UCSD). Introduction to F-isocrystals. I

Prof. Kedlaya kindly agreed to give two introductory lectures on this
important subject (the second one on March 4).

Abstract

Let X be a variety over a field of characteristic p>0. The notion of
l-adic local system on X has not one but two p-adic analogs, called
"convergent F-isocrystal" and "overconvergent F-isocrystal". I will start
from scratch and give an overview of the theory of F-isocrystals, paying
close attention to analogies from the l-adic case.



• Date: Mon, 15 Feb 2016 19:06:37

• Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202.
No meetings until then.

Kedlaya's notes on F-isocrystals are here:
http://kskedlaya.org/papers/isocrystals.pdf



• Date: Mon, 29 Feb 2016 19:34:06

• Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202.
(Unusual day, time, and room!)

Kedlaya's notes on F-isocrystals are updated:
http://kskedlaya.org/papers/isocrystals.pdf



• Date: Thu, 3 Mar 2016 12:32:11

• Kiran Kedlaya will speak tomorrow (Friday) at 4:00 p.m. in room E202.
(Unusual day, time, and room!)



• Date: Fri, 4 Mar 2016 18:40:50

• No seminar on Monday.

Thursday (March 10), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric
Langlands correspondence.

Abstract

The classical limit of the geometric Langlands correspondence is the
conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a
reductive group G and its dual. There was a significant progress on this
statement when G=GL(n) and the Hitchin fibers are identified with
compacitified Jacobians of spectral curves. Unfortunately, the methods are
specific to the group GL(n), and much less is known about the case of
general G.

In my talk, I plan to review the current state of the area, and then
sketch a new approach (work in progress joint with R.Fedorov). The
approach is based on studying the classical limit of the Hecke algebra',
which turns out to be a much richer object than its usual (quantum')
counterpart.



• Date: Mon, 7 Mar 2016 17:41:41

• Thursday (March 10), 4:30 p.m, room E 206.
Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric
Langlands correspondence.

Abstract

The classical limit of the geometric Langlands correspondence is the
conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a
reductive group G and its dual. There was a significant progress on this
statement when G=GL(n) and the Hitchin fibers are identified with
compacitified Jacobians of spectral curves. Unfortunately, the methods are
specific to the group GL(n), and much less is known about the case of
general G.

In my talk, I plan to review the current state of the area, and then
sketch a new approach (work in progress joint with R.Fedorov). The
approach is based on studying the classical limit of the Hecke algebra',
which turns out to be a much richer object than its usual (quantum')
counterpart.



• Date: Thu, 10 Mar 2016 18:38:11

• No more meetings this quarter.



• Date: Mon, 28 Mar 2016 15:17:01

• No meetings this week and the next one.

On April 11 (Monday)  Kazuya Kato will begin his series of talks.



• Date: Thu, 7 Apr 2016 17:30:30

• Monday (April 11), 4:30 p.m, room E 206.
Kazuya Kato. On compactifications of
the moduli spaces of Drinfeld modules. I.

Abstract

The main subject in my talk is to construct

(1) toroidal compactifications of the moduli spaces of Drinfeld modules.

This is similar to the well known

(2) toroidal compactifications of the moduli spaces of polarized abelian
vareities.

But there is a big difference as is explained below.

This is a joint work with T. Fukaya and R. Sharifi. We are studying the
analogue of Sharifi conjecture (it is a refined Iwasawa theory) for GL(n)
of a function field, and then the nice toroidal boundary of the moduli
space of Drinfeld modules became necessary.

The moduli space of polarized abelian varieties has two kinds of
compactification, Satake-Baily-Borel compactification and toroidal
compactifications. For the moduli space of Drinfeld modules, an
analogue of the Satake-Baily-Borel compactification was constructed by
Kapranov and Pink. The analogy is very strong here. Pink wrote a short
paper in 1994 on toroidal compactifications of the moduli space of
Drinfeld modules. But the details are not yet published.

For the toroidal compactification, there is a big difference between (1)
and (2). These toroidal compactifications treat degenerations of Drinfeld
modules and of polarized abelian varieties, respectively. In the
degeneration, the local monodromy of a degenerating polarized abelian
variety has length of unipotence two, but the local monodromy
of a degenerating Drinfeld module can have bigger length of unipotence. In
my talk,

1. I give an overview of the analytic theory and explain how such
difference of (1) and (2) appears.

2. I explain the theory of degeneration of Drinfeld modules (theory of
iterated Tate uniformizations, where the iteration is necessary to treat
the larger length of the unipotence).

To my regret, these 1 and 2 are the best things which I can explain well
now. We have not yet completed a paper on the construction of the toroidal
compactifications  and we are not yet perfectly sure that the proofs are
OK. I hope to explain the construction in a next opportunity.



• Date: Mon, 11 Apr 2016 19:31:45

• No seminar on Thursday.
Kazuya Kato will continue on Monday April 18.



• Date: Thu, 14 Apr 2016 17:09:21

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

Kazuya Kato. On compactifications of the
moduli spaces of Drinfeld modules. II.

Abstract

An elliptic curve over \C is presented as \C/L where L is a \Z-lattice  of
rank 2. Tate discovered that an elliptic curve over a local field K  has
the presentation K^\times/q^{\Z} as the quotient of the multiplicative
group of K by a \Z-lattice q^{\Z} of rank 1 if the  elliptic curve has
multiplicative reduction. This was generalized by  Raynaud, Mumford,
Faltings-Chai to higher dimensional abelian varieties  over the quotient
field of a complete local normal integral domain of  higher dimension.

In his first paper on Drinfeld modules, Drinfeld proved that the Drinfeld
module over a local filed K has a similar presentation K/L as  the
quotient of the additive group K by a certain lattice L.

I will explain how to generalize this theory of Drinfeld to the higher
dimensional base case.

This is a theory of degeneration of Drinfeld modules. This is important
for the toroidal compactification of the moduli space of Drinfeld
modules. The q of Tate in the case of an elliptic curve is the best
coordinate function of the compactified modular curve at the cusp, but q
is not an algebraic function. It is an analytic function or a function
which appears after the completion. The best coordinate functions of the
toroidal compactification at the boundary are not algebraic, and they
appear to classify the lattice L which appears after the completion.

There is a big difference from the case of abelian variety. This
difference is due to the fact that the length of the unipotence of the
local monodromy is two for abelian varieties but can be bigger for
Drinfeld modules.



• Date: Tue, 19 Apr 2016 10:39:42

• No seminar on Thursday April 21.



• Date: Mon, 25 Apr 2016 16:57:45

• Thursday (April 28), 4:30 p.m, room E 206.
Tsao-Hsien Chen (NWU) Quantization of Hitchin integrable system via
positive characteristic.

Abstract

In their work
"Quantization of Hitchin's integrable system and Hecke eigensheaves",
Beilinson and Drinfeld give a construction of an automorphic D-module
corresponding to a local system which carries an additional structure of
an oper.

In my talk, I will explain a new proof of this result, in the case of
G=GL(n), via positive characteristic method. This talk is based on joint
work with R.Bezrukavnikov, R.Travkin and X.Zhu.



• Date: Sun, 1 May 2016 10:52:15

• No seminar on Monday.

****

Thursday (May 5), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. I.

Abstract

In his recent article "The characteristic cycle and the singular support
of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the
theory of characteristic cycle that generalizes the theory of
Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties
over a field of arbitrary characteristic. For sheaves on a curve the
characteristic cycle amounts to the Artin conductor. One of the central
results of the theory is the global Euler characteristic formula; for a
curve this is the classical formula of Grothendieck-Ogg-Shafarevich.

In the talks I will explain the principal ideas of Saito's theory and
sketch the proofs. They continue my winter talks about the singular
support; all necessary facts will be reminded.



• Date: Wed, 4 May 2016 16:48:24

• Thursday (May 5), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. I.

Abstract

In his recent article "The characteristic cycle and the singular support
of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the
theory of characteristic cycle that generalizes the theory of
Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties
over a field of arbitrary characteristic. For sheaves on a curve the
characteristic cycle amounts to the Artin conductor. One of the central
results of the theory is the global Euler characteristic formula; for a
curve this is the classical formula of Grothendieck-Ogg-Shafarevich.

In the talks I will explain the principal ideas of Saito's theory and
sketch the proofs. They continue my winter talks about the singular
support; all necessary facts will be reminded.



• Date: Thu, 5 May 2016 19:07:42

• Presumably, Beilinson will continue on May 16.
Next week there will be two talks by Yun (Monday and Thursday).

*****

Monday (May 9), 4:30 p.m, room E 206.
Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of
L-functions for function fields. I.

Abstract

In joint work with Wei Zhang, we prove a higher derivative analogue of the
Waldspurger formula and the Gross-Zagier formula in the function field
setting under some unramifiedness assumptions. Our formula relates the
self-intersection number of certain cycles on the moduli of Drinfeld
Shtukas for GL(2) to higher derivatives of
automorphic L-functions for GL(2).

In the first talk I will give motivation and state the main results,
giving all the necessary definitions. In the second talk (Thursday May 12)
I will sketch the geometric ideas in the proof.



• Date: Mon, 9 May 2016 18:46:20

• Thursday (May 12), 4:30 p.m, room E 206.
Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of
L-functions for function fields. II.



• Date: Thu, 12 May 2016 18:35:10

• Monday (May 16), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. II.

Abstract

I will sketch the proofs of the theorems on characteristic cycles of etale
sheaves on varieties over a field of arbitrary characteristic.



• Date: Mon, 16 May 2016 19:21:00

• Thursday (May 19), 4:30 p.m, room E 206.
Alexander Beilinson. The characteristic cycle of an etale sheaf. III.



• Date: Thu, 19 May 2016 18:23:54

• Monday (May 23), 4:30 p.m, room E 206.
Sam Raskin (MIT). Single variable calculus and local geometric Langlands

Abstract

The moduli of (possibly irregular) formal connections in one variable (up
to gauge transformations) is an infinite-dimensional space that "feels
finite-dimensional," e.g., it has finite-dimensional tangent spaces.
However, it is not so clear how this perception is actually reflected in
the global geometry of this space.

Previous works have focused on explicit parametrization of this space. As
we will recall during the talk, this approach has significant limitations,
and is insufficient to say anything about the global geometry. But we will
instead find that this space appears kinder through the lens of
homological (or more poetically, noncommutative) geometry, exhibiting
better features than all its close relatives.

Finally, we will discuss how these results lend credence to the existence
of a de Rham Langlands program incorporating arbitrary singularities (the
usual story is unramified, or at worst has Iwahori ramification).



• Date: Mon, 23 May 2016 19:34:29

• Thursday (May 26), 4:30 p.m, room E 206.
Sam Raskin (MIT). W-algebras and Whittaker categories

Abstract

Affine W-algebras are a somewhat complicated family of (topological)
associative algebras associated with a semisimple Lie algebra, quantizing
functions on the algebraic loop space of Kostant's slice. They have
attracted a great deal of attention because of Feigin-Frenkel's duality
theorem for them, which identifies W-algebras for a Lie algebra and for
its Langlands dual through a subtle construction.

The purpose of this talk is threefold: 1) to introduce a natural family of
(discrete) modules for the affine W-algebra, 2) to prove an analogue of
Skryabin's equivalence in this setting, realizing the category of
(discrete) modules over the W-algebra in a more natural way, and 3) to
explain how these constructions help understand Whittaker categories in
the more general setting of local geometric Langlands. These three points
all rest on a simple geometric observation, which provides a family of
affine analogues of Bezrukavnikov-Braverman-Mirkovic.



• Date: Thu, 26 May 2016 19:21:41

• No more meetings of the seminar this quarter.



• Date: Mon, 26 Sep 2016 08:47:40

• No meetings this week.

The first meeting is on Oct. 3 (Monday), 4:30 p.m, room E 206.

I will discuss some recent results by K.Kedlaya and me, see
http://arxiv.org/abs/1604.00660

Let X be a variety over F_p . Fix a prime \ell different from p, an
algebraic closure \overline{Q_\ell}, and a p-adic valuation v of the
subfield of algebraic numbers in \overline{Q_\ell} normalized so that
v(p)=1. Let M be a \overline{Q_\ell}-sheaf on X such that for every closed
point x in X the eigenvalues of the geometric Frobenius acting on M_x are
algebraic numbers. Applying the valuation v to these numbers and dividing
by the degree of v over F_p, we get a collection of rational numbers,
which are called slopes of M at x.

We proved that if X is a smooth curve and M is an irreducible local system
then for almost all x the gaps between two consecutive slopes of M at x
are not greater than 1. If M has rank 2 then one can even replace “almost
all” by “all”, but in the rank 3 case this is false.

I will say almost nothing about the proof (which is based on the theory of
F-isocrystals).

`