This is an archive of email messages concerning the Geometric Langlands Seminar for 09-10.

No seminar on Thursday October 1 and Monday October 5.

The seminar will meet on October 8.


October 8, 4:30 p.m, room E 206.

Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.

                  Abstract

Let C be a (smooth projective algebraic) curve, in other words, a Riemann
surface. It is well known that the Jacobian J of C is a self-dual complex
torus, that is, J is identified with the space of topologically trivial
line bundles on J.

Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer
compact. By considering torsion-free sheaves instead of line bundles, one
obtains a natural singular compactification J' of J.

The subject of this talk is line bundles on J' and their cohomology. The
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the
autoduality.

My next talk will cover the stronger `compactified' autoduality statement
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.

The compactified Jacobians play a role in the geometric Langlands
correspondence (for GL(n)), where they appear as fibers of the Hitchin
fibration. However, the talk relies on classical methods of algebraic
geometry, and should be accessible to wide audience.

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

Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.

                  Abstract

Let C be a (smooth projective algebraic) curve, in other words, a Riemann 
surface. It is well known that the Jacobian J of C is a self-dual complex 
torus, that is, J is identified with the space of topologically trivial 
line bundles on J.

Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer 
compact. By considering torsion-free sheaves instead of line bundles, one 
obtains a natural singular compactification J' of J.

The subject of this talk is line bundles on J' and their cohomology. The 
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line 
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the 
autoduality.

My next talk will cover the stronger `compactified' autoduality statement 
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.

The compactified Jacobians play a role in the geometric Langlands 
correspondence (for GL(n)), where they appear as fibers of the Hitchin 
fibration. However, the talk relies on classical methods of algebraic 
geometry, and should be accessible to wide audience.

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

Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.

                  Abstract

Let C be a (smooth projective algebraic) curve, in other words, a Riemann
surface. It is well known that the Jacobian J of C is a self-dual complex
torus, that is, J is identified with the space of topologically trivial
line bundles on J.

Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer
compact. By considering torsion-free sheaves instead of line bundles, one
obtains a natural singular compactification J' of J.

The subject of this talk is line bundles on J' and their cohomology. The
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the
autoduality.

My next talk will cover the stronger `compactified' autoduality statement
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.

The compactified Jacobians play a role in the geometric Langlands
correspondence (for GL(n)), where they appear as fibers of the Hitchin
fibration. However, the talk relies on classical methods of algebraic
geometry, and should be accessible to wide audience.

There will be no meeting of the Langlands seminar on Monday.

Dima Arinkin will continue his talk on
   Autoduality for Jacobians of singular curves
on WEDNESDAY (Oct 14) at 4 p.m. in E 203
and then on Thursday (October 15) at 4:30 p.m in room E 206.

(Note that the algebraic geometry will not meet this Wednesday.)

I strongly recommend the students to really understand Dima's first talk.
I think this is doable.

You can download Dima Arinkin's article from
http://arxiv.org/abs/0705.0190

As far as I understand, the relevant parts of Mumford's book on
Abelian varieties are:
 Chapter II, Section 8
 and Chapter III, Section 13.

The next seminar is on
WEDNESDAY (Oct 14) at 4 p.m. in E 203
(please notice the unusual time and place).

Dima Arinkin will continue his talk on
   autoduality for Jacobians of singular curves

Tomorrow (Thursday) at 4:30 p.m in room E 206
Dima Arinkin will continue his talk on
   Autoduality for Jacobians of singular curves.

Because of the Albert lecture,
the Langlands seminar will not meet on Monday, October 19.

No seminar on Thursday, October 22.

Next Monday (October 26) David Nadler (NWU) will start his series of talks
 joint with John Francis.


Title of the series: Drinfeld centers and derived algebraic geometry.

Abstract: In this series of talks, we would like to describe the
background and results of our joint paper with David Ben-Zvi
"Integral transforms and Drinfeld centers in derived algebraic geometry",
http://arxiv.org/abs/0805.0157

No preliminary knowledge of derived algebraic geometry is assumed.

Our broad goal is to understand some of the basic homotopical algebra of
the tensor category of quasicoherent sheaves on a stack. More
specifically, we would like to understand how geometric operations on
stacks (fiber products, loop spaces,...) interact with algebraic
operations on categories of sheaves (tensor products, Hochschild
homology,...).

Our plan roughly breaks into three parts:

1. The first talk will be an example-oriented overview of the phenomena to
be studied. We will informally introduce the main objects and try to build
up enough intuition so the audience will have a good sense
as to what should be true.

2. In the second part, we will explain some of the general constructions
which we will later apply to categories of sheaves. We will follow Lurie's
foundations on algebra in the context of oo-categories. Our focus will be
on the notion of center, and in particular the examples of Drinfeld and 
E_n-centers of monoidal oo-categories.

3. In the third part, we will specialize to derived algebraic geometry. Here
is a sample of the kind of theorem we will discuss:

Theorem: For X a perfect stack (a notion which in characteristic zero
includes all of the commonly occurring examples in geometric
representation theory), the Drinfeld center of the monoidal category
QCoh(X) of quasicoherent sheaves on X is equivalent to the category
QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).

Monday (October 26), 4:30 p.m, room E 206.
David Nadler (NWU). Drinfeld centers and derived algebraic geometry. I.

                     Abstract

In this series of talks John Francis and I would like to describe the
background and results of our joint paper with David Ben-Zvi
"Integral transforms and Drinfeld centers in derived algebraic geometry",
http://arxiv.org/abs/0805.0157

No preliminary knowledge of derived algebraic geometry is assumed.

Our broad goal is to understand some of the basic homotopical algebra of
the tensor category of quasicoherent sheaves on a stack. More
specifically, we would like to understand how geometric operations on
stacks (fiber products, loop spaces,...) interact with algebraic
operations on categories of sheaves (tensor products, Hochschild
homology,...).

Our plan roughly breaks into three parts:

1. The first talk will be an example-oriented overview of the phenomena to
be studied. We will informally introduce the main objects and try to build
up enough intuition so the audience will have a good sense
as to what should be true.

2. In the second part, we will explain some of the general constructions
which we will later apply to categories of sheaves. We will follow Lurie's
foundations on algebra in the context of oo-categories. Our focus will be
on the notion of center, and in particular the examples of Drinfeld and 
E_n-centers of monoidal oo-categories.

3. In the third part, we will specialize to derived algebraic geometry.
Here is a sample of the kind of theorem we will discuss:

Theorem: For X a perfect stack (a notion which in characteristic zero
includes all of the commonly occurring examples in geometric
representation theory), the Drinfeld center of the monoidal category
QCoh(X) of quasicoherent sheaves on X is equivalent to the category
QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).

No seminar on Thursday.

The next talk on "Drinfeld centers and derived algebraic geometry" will be
given on Monday (November 2) by John Francis (NWU).

Monday (November 2), 4:30 p.m, room E 206.
John Francis (NWU). Drinfeld centers and derived algebraic geometry. II.

>                      Abstract
>
> In this series of talks John Francis and I would like to describe the
> background and results of our joint paper with David Ben-Zvi
> "Integral transforms and Drinfeld centers in derived algebraic geometry",
> http://arxiv.org/abs/0805.0157
>
> No preliminary knowledge of derived algebraic geometry is assumed.
>
> Our broad goal is to understand some of the basic homotopical algebra of
> the tensor category of quasicoherent sheaves on a stack. More
> specifically, we would like to understand how geometric operations on
> stacks (fiber products, loop spaces,...) interact with algebraic
> operations on categories of sheaves (tensor products, Hochschild
> homology,...).
>
> Our plan roughly breaks into three parts:
>
> 1. The first talk will be an example-oriented overview of the phenomena to
> be studied. We will informally introduce the main objects and try to build
> up enough intuition so the audience will have a good sense
> as to what should be true.
>
> 2. In the second part, we will explain some of the general constructions
> which we will later apply to categories of sheaves. We will follow Lurie's
> foundations on algebra in the context of oo-categories. Our focus will be
> on the notion of center, and in particular the examples of Drinfeld and
> E_n-centers of monoidal oo-categories.
>
> 3. In the third part, we will specialize to derived algebraic geometry.
> Here is a sample of the kind of theorem we will discuss:
>
> Theorem: For X a perfect stack (a notion which in characteristic zero
> includes all of the commonly occurring examples in geometric
> representation theory), the Drinfeld center of the monoidal category
> QCoh(X) of quasicoherent sheaves on X is equivalent to the category
> QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).

We do not meet tomorrow (Monday) because the speaker is sick.
Presumably we meet on Thursday (November 5).

> John Francis (NWU). Drinfeld centers and derived algebraic geometry. II.
>
>>                      Abstract
>>
>> In this series of talks John Francis and I would like to describe the
>> background and results of our joint paper with David Ben-Zvi
>> "Integral transforms and Drinfeld centers in derived algebraic
>> geometry",
>> http://arxiv.org/abs/0805.0157
>>
>> No preliminary knowledge of derived algebraic geometry is assumed.
>>
>> Our broad goal is to understand some of the basic homotopical algebra of
>> the tensor category of quasicoherent sheaves on a stack. More
>> specifically, we would like to understand how geometric operations on
>> stacks (fiber products, loop spaces,...) interact with algebraic
>> operations on categories of sheaves (tensor products, Hochschild
>> homology,...).
>>
>> Our plan roughly breaks into three parts:
>>
>> 1. The first talk will be an example-oriented overview of the phenomena
>> to
>> be studied. We will informally introduce the main objects and try to
>> build
>> up enough intuition so the audience will have a good sense
>> as to what should be true.
>>
>> 2. In the second part, we will explain some of the general constructions
>> which we will later apply to categories of sheaves. We will follow
>> Lurie's
>> foundations on algebra in the context of oo-categories. Our focus will
>> be
>> on the notion of center, and in particular the examples of Drinfeld and
>> E_n-centers of monoidal oo-categories.
>>
>> 3. In the third part, we will specialize to derived algebraic geometry.
>> Here is a sample of the kind of theorem we will discuss:
>>
>> Theorem: For X a perfect stack (a notion which in characteristic zero
>> includes all of the commonly occurring examples in geometric
>> representation theory), the Drinfeld center of the monoidal category
>> QCoh(X) of quasicoherent sheaves on X is equivalent to the category
>> QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
>

Thursday (November 5), 4:30 p.m, room E 206.
John Francis (NWU) will give the second talk in the series
"Drinfeld centers and derived algebraic geometry".
As far as I understand, it will be independent from the first one
(which was given last Monday by David Nadler).

                     Abstract

This talk will give an exposition of E_n centers, where E_n is the operad
of configuration spaces of n-disks. The main case of interest will be the
E_n or Drinfeld (for n=1) center of a monoidal
oo-category. We will preface this topic with the necessary review of the
theory of oo-categories (definition, limits and colimits,
stability, presentability, operads and operadic monoidal structures) and,
in particular, a description of oo-categories of functors between
oo-categories.

Monday (November 9), 4:30 p.m, room E 206.
David Ben-Zvi (University of Texas at Austin).
Drinfeld centers and derived algebraic geometry. III.

                     Abstract

In this talk we will discuss the interaction between geometric operations on
schemes and stacks and algebraic operations on their categories of sheaves
in the context of derived algebraic geometry. We will introduce derived
schemes and stacks and their oo-categories of quasicoherent sheaves. The
latter can be considered as plain, monoidal, E_n or symmetric monoidal
oo-categories, leading to a collection of natural algebraic operations to
perform on them. We will demonstrate how the results of these
operations can be described themselves as oo-categories of quasicoherent
sheaves. As an application we discuss (time permitting) the representation
theory of quasicoherent Hecke categories.

The prerequisites are the basic notions of oo-categories, as presented in
John Francis' talk, or a willingness to work through formal analogies,
such as those presented in David Nadler's talk.

Thursday (November 12), 4:30 p.m, room E 206.
Ivan Losev (MIT) Irreducible finite-dimensional
representations for finite W-algebras.

                     Abstract
(Finite) W-algebras are certain associative
algebras arising in Lie theory. They are constructed
from a pair of a semisimple Lie algebra and its
nilpotent element. They originate from Kostant's
paper "On Whittaker vectors and representation theory",
where the case of a principal nilpotent element
was considered. During the last decade they attracted
some interest from representation theorists (Brundan,
Ginzburg, Goodwin, Kleshchev, Premet, the speaker
and others) mainly due to their relation to representation
theory of universal enveloping algebras, both
in zero and positive characteristics. In my talk
I will describe known results and conjectures on the
classification of finite dimensional irreducible
modules for W-algebras.

References: arXiv:0707.3108, 0807.1023, 0812.1584.

Thursday (November 12), 4:30 p.m, room E 206.
Ivan Losev (MIT) Irreducible finite-dimensional
representations for finite W-algebras.

                     Abstract
(Finite) W-algebras are certain associative
algebras arising in Lie theory. They are constructed
from a pair of a semisimple Lie algebra and its
nilpotent element. They originate from Kostant's
paper "On Whittaker vectors and representation theory",
where the case of a principal nilpotent element
was considered. During the last decade they attracted
some interest from representation theorists (Brundan,
Ginzburg, Goodwin, Kleshchev, Premet, the speaker
and others) mainly due to their relation to representation
theory of universal enveloping algebras, both
in zero and positive characteristics. In my talk
I will describe known results and conjectures on the
classification of finite dimensional irreducible
modules for W-algebras.

References: arXiv:0707.3108, 0807.1023, 0812.1584.

No seminar on Monday (November 16).

Roman Bezrukavnikov will speak on Thursday (November 19).

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

Roman Bezrukavnikov (MIT).
Character D-modules as center of the categorical Hecke algebra.

                     Abstract

Lusztig introduced character sheaves on a reductive algebraic
group as a tool for understanding characters of finite Chevalley groups.
In a joint work with Finkelberg and Ostrik we show how to view the 
corresponding objects in the D-module setting as objects of the
categorical center of the categorical Hecke algebra and rederive Lusztig's

classification results through this approach.
Some of our results are related to (though different from) results of 
Ben-Zvi, Francis and Nadler.

No more Langlands seminar this quarter.

Happy New Year!

Here is a tentative program for the winter quarter.

First, Sasha Beilinson will give several talks on motive theory.
The title of his series is "Finite-dimensionality of motives and some
problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
As far as I know, Sasha will start from scratch and discuss some
fundamental open problems, whose formulation is very easy to understand.
He will begin either on January 7 or more likely, on January 11.

Yakov Varshavsky will give several talks  on his generalization of
Fujiwara's theorem. Presumably he will begin on January 25.

In February Dmitry Kaledin will give a series of talks on his proof of
noncommutative Hodge-to-de Rham degeneration using a noncommutative
version of the Deligne-Illusie method. Presumably he will begin on
February 8.

No seminar on Thursday (January 7).

On January 11 (Monday) Sasha Beilinson will give the first talk in his
series on motives. His talks will be introductory;
no initial knowledge about motives is required.

Title: "Finite-dimensionality of motives and some
problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".

          Abstract

The theory of motives predicts a number of properties of
the groups of algebraic cycles. One needs them in order to be able
to use motives as effectively as usual homology groups.
I will mostly discuss the finite-dimensionality property of
motives, conjectured by Kimura and O'Sullivan, and relateted
problems about algebraic cycles.

The talks are introductory; no initial knowledge about
motives is required. One of my goals is to explain
HOW LITTLE WE KNOW about algebraic cycles.

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

Alexander Beilinson.
Title: "Finite-dimensionality of motives and some
problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".

          Abstract

The theory of motives predicts a number of properties of
the groups of algebraic cycles. One needs them in order to be able to use
motives as effectively as usual homology groups.
I will mostly discuss the finite-dimensionality property of
motives, conjectured by Kimura and O'Sullivan, and relateted
problems about algebraic cycles.

The talks are introductory; no initial knowledge about
motives is required. One of my goals is to explain
HOW LITTLE WE KNOW about algebraic cycles.

This is just to remind that Beilinson begins his series of talks
today and that this series can be easily understood by beginners.

> Monday (January 11), 4:30 p.m, room E 206.
>
> Alexander Beilinson.
> Title: "Finite-dimensionality of motives and some
> problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
>
>           Abstract
>
> The theory of motives predicts a number of properties of
> the groups of algebraic cycles. One needs them in order to be able to use
> motives as effectively as usual homology groups.
> I will mostly discuss the finite-dimensionality property of
> motives, conjectured by Kimura and O'Sullivan, and relateted
> problems about algebraic cycles.
>
> The talks are introductory; no initial knowledge about
> motives is required. One of my goals is to explain
> HOW LITTLE WE KNOW about algebraic cycles.
>
>
>
>
>
>
>
>
>
>

No seminar on Thursday.
Beilinson will continue on January 18 (Monday).

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

Alexander Beilinson will give his second talk on motives.
It will be more or less independent of the first one.
(But of course, understand the first talk is worth trying;
in particular, this would help you understand the second one.)

> Title: "Finite-dimensionality of motives and some
> problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
>
>           Abstract
>
> The theory of motives predicts a number of properties of
> the groups of algebraic cycles. One needs them in order
> to be able to use
> motives as effectively as usual homology groups.
> I will mostly discuss the finite-dimensionality property of
> motives, conjectured by Kimura and O'Sullivan, and relateted
> problems about algebraic cycles.
>
> The talks are introductory; no initial knowledge about
> motives is required. One of my goals is to explain
> HOW LITTLE WE KNOW about algebraic cycles.

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

Alexander Beilinson will give his last talk on motives.

Beilinson's notes of his second and third talks on motives.are attached.

Next talk:
Monday (January 25), 4:30 p.m, room E 206.

Yakov Varshavsky (Hebrew University)
Lefschetz-Verdier trace formula, contracting correspondences, and
a simple proof of Fujiwara's theorem.

                              Abstract

A theorem of Fujiwara (formerly Deligne's conjecture) asserts that the
Lefschetz-Verdier trace formula has a particularly simple and explicit
form in the case when a correspondence is defined over a finite field and
is twisted by a sufficiently large power of Frobenius.

In my talks I will indicate a simple proof of this theorem based on an
algebro-geometric notion of a contracting correspondence and a classical
construction of a deformation to the normal cone.

The talk is based on the following articles (the first one is a research
announcement, the second one is the full version):

<http://front.math.ucdavis.edu/0505.5314>
"A proof of a generalization of Deligne's conjecture"

<http://front.math.ucdavis.edu/0505.5564>
"Lefschetz-Verdier trace formula and a generalization
of a theorem of Fujiwara"

Attachment: Sasha_on_motives.pdf
Description: Adobe PDF document

Yakov Varshavsky will continue his talk on
Thursday (January 28), 4:30 p.m, room E 206.

> Yakov Varshavsky (Hebrew University)
> Lefschetz-Verdier trace formula, contracting correspondences, and
> a simple proof of Fujiwara's theorem.
>
>                               Abstract
>
> A theorem of Fujiwara (formerly Deligne's conjecture) asserts that the
> Lefschetz-Verdier trace formula has a particularly simple and explicit
> form in the case when a correspondence is defined over a finite field and
> is twisted by a sufficiently large power of Frobenius.
>
> In my talks I will indicate a simple proof of this theorem based on an
> algebro-geometric notion of a contracting correspondence and a classical
> construction of a deformation to the normal cone.
>
> The talk is based on the following articles (the first one is a research
> announcement, the second one is the full version):
>
> <http://front.math.ucdavis.edu/0505.5314>
> "A proof of a generalization of Deligne's conjecture"
>
> <http://front.math.ucdavis.edu/0505.5564>
> "Lefschetz-Verdier trace formula and a generalization
> of a theorem of Fujiwara"
>
>
>
>
>
>
>

No seminar on Monday (February 1).

On Thursday (February 4) Dmitry Kaledin will begin his series of talks.
(The goal of this series is to explain his proof of
the degeneration of the Hodge-to-de Rham spectral sequence in
noncommutative algebraic geometry.)

Thursday (February 4), 4:30 p.m, room E 206.

Dmitry Kaledin (Moscow) will give the first talk in his series
on the Hodge-to-de Rham degeneration theorem in
noncommutative algebraic geometry. In particular, he will explain the
setting of noncommutative algebraic geometry.

   Title of the talk:
Finiteness conditions for DG algebras.

                   Abstract

I am going to discuss some general facts about DG categories and DG
algebras -- this is what constitutes the "non-commutative setting" of the
theorem -- and then I am going to concentrate on a beautiful recent
theorem of Bertrand Toen which claims that a proper and smooth DG algebra
over a field K actually comes from a proper and smooth DG algebra over a
finitely generated subring in K.

1. Here is a reference to Toen's article discussed in Kaledin's talk:

MR2441136 (2009g:16017)
Bertrand Toen, Anneaux de definition des dg-algebres propres et lisses.
Bull. Lond. Math. Soc. 40 (2008), no. 4, 642--650.

2. Kaledin's next talk:

Monday (February 8), 4:30 p.m, room E 206.

Title: Non-commutative Hodge-to-de Rham degeneration I.

                     Abstract

In this lecture, I am going to explain two things:
(1) the beautiful short proof of Hodge-to-de Rham degeneration
in the usual commutative setting given by Deligne-Illusie, and all it
requires (which is basically the Cartier isomorphism in char p).
(2) non-commutative generalizations of differential forms
and de Rham cohomology classes, and the statement of the
Hodge-to-de Rham degeneration theorem in the non-commutative case.

How (1) is related to (2) is the subject of subsequent lectures.

I plan to go very slowly; in particular, I will assume no knowledge
whatsoever about de Rham cohomology in positive characteristic.

1. Kaledin's next talk:

Thursday (February 11), 4:30 p.m, room E 206.


Title: Hodge-to-de Rham degeneration II

Abstract. In this lecture, I will construct a version of Cartier
isomorphism for an associative algebra A over a finite field k; A
has to satisfy some assumptions analogous to the ones we had in the
commutative case (dim X < p, X liftable to W_2(k)). Then if time
permits, I will sketch the proof of the degeneration theorem.
=======================================================================

2. Here are the links mentioned in Kaledin's last talk.
His Tokyo lectures are here:

http://imperium.lenin.ru/~kaledin/tokyo/

There are also his lectures from last year from Seoul, on a related
subject (in particular, in lecture 1 there is sketch of finiteness results
for DG algebras which he explained last time):

http://imperium.lenin.ru/~kaledin/seoul/

As for the degeneration theorem itself, the main paper is
arXiv:math/0611623, and there is an overview in arXiv:0708.1574.