This is an archive of email messages concerning the Geometric Langlands Seminar for 07-08.

As usual, the seminar will meet on Mondays and/or Thursdays in room E206 
from 4:30 p.m. until the time when we get tired (e.g., until 7 or 7:30
p.m.).

This quarter we begin with a series of talks on modular categories.
Namely, on October 4 and 8 Dmitry Nikshych will speak on his recent works.
On the other hand, Mitya Boyarchenko will give
three introductory lectures on this subject,
of which two will be given before Nikshych's talks
(namely, on Thursday Sep 27 and Monday Oct 1).
Beginners (e.g., second year graduate students)
are strongly recommended to attend
at least, the introductory lectures.

Modular categories are closely related to low-dimensional topology and
conformal field theory.
But why should a person working in geometric representation theory be
interested in modular categories? Here is a reason.
A typical example of a modular category is the category of
sheaves of vector spaces on a finite group G equivariant with respect to
G-conjugation. This category has an important generalization,
namely the (derived) category of constructible sheaves on an algebraic
group G. It is believed (and in certain cases, proved) that such a
category can be decomposed, in some sense, as a direct integral of modular
categories
(as far as I understand, this is what Lusztig's theory of character
sheaves is about).

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

Mitya Boyarchenko will give his first introductory lecture on modular
categories. He will begin from scratch, so
second year students are strongly recommended to attend.


                       Abstract


The goal of my series of talks is to give an introduction
to the theory of modular categories and explain their
relationship to the notion of a modular functor in
topological field theory.

In the first lecture I will review the necessary background
from the theory of monoidal categories and braided monoidal
categories. In particular, I will recall the notions of a
braided monoidal category, a fusion category, and a ribbon
category. Time permitting, I will describe the "graphical
calculus" for morphisms in a ribbon category.

In the second lecture I will define the notion of a modular
category and give several different (algebraic) equivalent
characterizations of modular categories. I will also
explain some elementary examples of modular categories.

The first two lectures will be purely algebraic; the
relation with topology of surfaces and 3-manifolds will
be discussed when my series of lectures continues
after the talks that will be given by Nikshych.

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

Mitya Boyarchenko will give his first introductory lecture on modular
categories. He will begin from scratch, so
second year students are strongly recommended to attend.


                       Abstract


The goal of my series of talks is to give an introduction
to the theory of modular categories and explain their
relationship to the notion of a modular functor in
topological field theory.

In the first lecture I will review the necessary background
from the theory of monoidal categories and braided monoidal
categories. In particular, I will recall the notions of a
braided monoidal category, a fusion category, and a ribbon
category. Time permitting, I will describe the "graphical
calculus" for morphisms in a ribbon category.

In the second lecture I will define the notion of a modular
category and give several different (algebraic) equivalent
characterizations of modular categories. I will also
explain some elementary examples of modular categories.

The first two lectures will be purely algebraic; the
relation with topology of surfaces and 3-manifolds will
be discussed when my series of lectures continues
after the talks that will be given by Nikshych.

Mitya Boyarchenko's notes for the lecture he gave
today are now posted online at the following URL:

http://www.math.uchicago.edu/~mitya/modular/modular1.pdf

Note that the notes cover more material than was
mentioned today. Some of the material covered in
the notes will be explained in the next lecture,
notably, the notions of pivotal/spherical and
tensor/fusion categories.

Mitya will also hold office hours for his lecture,
starting at 3pm this Sunday, in Eckhart 206.
Everyone is welcome. However, those who plan to
attend these office hours are encouraged to look
over Mitya's notes first, and prepare questions.

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

Mitya Boyarchenko will give his second introductory lecture on modular
categories.

                       Abstract


In the first lecture we mainly discussed rigid monoidal
categories, which are monoidal categories satisfying a
certain property. In order to define the notion of a
modular category, which will be the ultimate goal of
the next lecture, we need to introduce two additional
structures that can be put on a monoidal category,
namely, a pivotal structure, and a braiding.

These notions, together with the notions of a ribbon
category and that of a modular category, are the subject
of the second lecture. A brief summary of what I hope
to cover is as follows.

The informal meaning of a pivotal structure on a rigid
monoidal category M is that it is something which allows
one to give a natural definition of the trace of an
endomorphism of any object of M.

On the other hand, a braiding on M is a choice of
functorial isomorphisms between X\otimes Y and
Y\otimes X for every pair of objects X and Y of M,
satisfying certain natural identities.

A rigid monoidal category equipped with a pivotal
structure and a braiding satisfying a certain
compatibility condition is called a ribbon category.
Finally, a modular category is a ribbon category
satisfying some finiteness conditions, and having
a certain nondegeneracy property.

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

Dmitri Nikshych (University of New Hampshire)
Morita duality for fusion categories and geometry of modular
categories.

                        Abstract

By a fusion category we mean an abelian semisimple tensor category
satisfying certain  finiteness conditions. The notion of a module
category over a fusion category can be viewed as a categorification of the
notion  of a module over a ring. It leads to a categorical Morita duality.
We will discuss these notions and will explain how the
above duality can be studied using modular categories.

A good deal of intuition about modular categories comes from metric Lie
algebras (i.e., Lie algebras equipped with a non-degenerate invariant 
scalar product). We will explain how Mueger's theory of centralizers in
modular categories makes it possible to extend some of the classical
linear algebra results to a categorical setting (this is based on a joint
work with V. Drinfeld, S. Gelaki, and V.Ostrik)

We will also discuss a joint work with D. Naidu about Morita equivalence
of group-theoretical fusion categories and braided equivalence of twisted
doubles of finite groups.

Mitya Boyarchenko's notes for his second talk on modular
categories have been posted at the following URL:

http://www.math.uchicago.edu/~mitya/modular/modular2.pdf

They cover most of what was explained during the lecture.
In addition, a portion of the notes (pages 6-13) is devoted
to a discussion of some foundational material, namely, the
notions of a monoidal category and of a strict monoidal
category; the notions of a weak/strong/strict monoidal
functor; and the notion of a morphism of monoidal functors.

The notion of a monoidal functor is important (in particular,
becomes more than just a "technicality) in the theory of
module categories over monoidal categories, which will be
one of the main themes in Nikshych's talks.

Unfortunately, Mitya's notes for the second lecture omit
several important topics, notably, the definition of the
Drinfeld center of a monoidal category and the definition
of the category of equivariant sheaves on a finite group
(they were briefly mentioned during the lecture). For the
first notion we recommend section XIII.4 in Kassel's book
"Quantum groups" as a reference.

1. The Geometric Langlands seminar webpage now has a new location:

http://www.math.uchicago.edu/~mitya/langlands.html

It will be maintained by Mitya Boyarchenko from now on.
A few of the links on the webpage have been updated.

2. Tomorrow talk:

> October 4 (Thursday), 4:30 p.m, room E 206.
>
> Dmitri Nikshych (University of New Hampshire)
> Morita duality for fusion categories and geometry of modular
> categories.
>
>                         Abstract
>
> By a fusion category we mean an abelian semisimple tensor category
satisfying certain  finiteness conditions. The notion of a module
category over a fusion category can be viewed as a categorification of
the
> notion  of a module over a ring. It leads to a categorical Morita
duality.
> We will discuss these notions and will explain how the
> above duality can be studied using modular categories.
>
> A good deal of intuition about modular categories comes from metric Lie
algebras (i.e., Lie algebras equipped with a non-degenerate invariant
scalar product). We will explain how Mueger's theory of centralizers in
modular categories makes it possible to extend some of the classical
linear algebra results to a categorical setting (this is based on a
joint
> work with V. Drinfeld, S. Gelaki, and V.Ostrik)
>
> We will also discuss a joint work with D. Naidu about Morita equivalence
of group-theoretical fusion categories and braided equivalence of
twisted
> doubles of finite groups.
>
>
>

1.Mitya Boyarchenko's notes of Nikshych's first talk
are available at the following URL:
http://www.math.uchicago.edu/~mitya/modular/nikshych1.pdf

2. Dmitri Nikshych will give his second talk on
Monday (October 8), 4:30 p.m, room E 206.

Title: Morita duality for fusion categories and geometry
of modular categories II.

                  Abstract

This will be a continuation of the talk given
last  Thursday. We will show that two fusion categories
are Morita  equivalent if and only if their centers are
equivalent  as braided fusion categories.

Recall that the center of a spherical fusion category
is a  modular category. Thus, to study spherical fusion
categories  up to a Morita equivalence is the same thing
as to study symmetries of modular categories.
As a first attempt of such a study we will explain how
to describe braided equivalences between a given
modular category C and the centers of categories of
group-graded vector spaces in terms of Lagrangian
subcategories of C. We will also explain how to determine
when two categories of group-graded vector spaces
(for possibly different groups) have braided equivalent
centers.

If time allows, we will introduce nilpotent fusion categories
(which generalize group-graded vector spaces in the same way
nilpotent groups generalize abelian groups) and discuss how
to apply the above ideas to them.

The material of this talk is based on joint works with
V. Drinfeld, S. Gelaki, D. Naidu, and V. Ostrik.

1. Mitya Boyarchenko's notes of Nikshych's today's lecture is
available at the following URL:

http://www.math.uchicago.edu/~mitya/modular/nikshych2.pdf

2. There will be no seminar on Thursday.

Monday (October 15), 4:30 p.m, room E 206.

Mitya Boyarchenko. Modular categories and modular functors.


                  Abstract

I will explain one of the several rigorous interpretations of the
(imprecise) statement that "a modular category is the same thing
as a 2-dimensional topological modular functor". I will also
explicitly describe the modular functors giving rise to the
examples of modular categories described in the earlier lectures.

The presentation will closely follow the paper "On the
Lego-Teichmuller game" of Bakalov and Kirillov (and chapter 5
of their book). If time permits, we will say a few words about
the relationship between their approach and the notion of a
2-dimensional modular functor studied in Turaev's book.

Monday (October 15), 4:30 p.m, room E 206.

Mitya Boyarchenko. Modular categories and modular functors.


                  Abstract


I will explain one of the several rigorous interpretations of the
(imprecise) statement that "a modular category is the same thing
as a 2-dimensional topological modular functor". I will also
explicitly describe the modular functors giving rise to the
examples of modular categories described in the earlier lectures.

The presentation will closely follow the paper "On the
Lego-Teichmuller game" of Bakalov and Kirillov (and chapter 5
of their book). If time permits, we will say a few words about
the relationship between their approach and the notion of a
2-dimensional modular functor studied in Turaev's book.

Mitya Boyarchenko asked me to update the announcement
of his lecture tomorrow. (The announcement that was sent out
previously remains in effect for his NEXT lecture.)


Monday (October 15), 4:30 p.m, room E 206.
Mitya Boyarchenko. Introduction to modular categories.III.

                  Abstract

My lecture on Monday will be devoted to some algebraic preparations
that are needed to study the relationship between modular categories
and modular functors. In particular, I will begin by explaining a
somewhat more natural and understandable definition of a modular
category. I will then recall the definitions of the s-matrix and
the central charge of a modular category, and explain how a modular
category gives rise to a projective finite dimensional representation
of the group SL_2(Z), which is the reason for the term "modular".

The lecture will be rather independent from my first two lectures.
In order to follow it one only needs to be familiar with the notions
of a rigid monoidal category and of a braided monoidal category.
These notions are discussed in some detail in the notes for my first
two lectures, which are available online:

http://www.math.uchicago.edu/~mitya/modular/modular1.pdf
http://www.math.uchicago.edu/~mitya/modular/modular2.pdf

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

Mitya Boyarchenko. Modular categories and modular functors.


                   Abstract


I will explain one of the several rigorous interpretations of
the (imprecise) statement that "a modular category is the same
thing as a 2-dimensional topological modular functor".

Due to time constraints, I will probably have to restrict
attention to modular categories whose multiplicative central
charge (which was defined in the last lecture) is equal to 1.
Such a category gives rise to a 2-dimensional topological
modular functor in the usual sense of this term (which I will
explain), and, in particular, to a finite dimensional
representation on the mapping class group M_g for every
g=1,2,...; for g=1, this representation is the same as the
representation of SL_2(Z) described in the previous lecture.

The presentation will closely follow Chapter 5 of the book
"Lectures on Tensor Categories and Modular Functors" by
Bakalov and Kirillov.

No Langlands seminar on Oct 22 (Monday).
Alexander Braverman will speak on Oct 25 (Thursday).

Mitya Boyarchenko's notes of his todays lecture are available at
http://www.math.uchicago.edu/~mitya/modular/modular4.pdf

October 25 (Thursday), 4:30 p.m, room E 206.
Alexander Braverman (Brown University).
Hecke operators for affine Kac-Moody groups

                  Abstract

The starting point for the Langlands conjecutres is the Satake isomorphism
which describes the spherical Hecke algebra of a sprit redusctive
p-adic group G in terms of the representation ring of its Langlands dual 
group. In this talk we shall generalize this result to the case when G is
an affine Kac-Moody group. Depending on how much time is left the 
following topics may also be discussed:
- Hecke operators and Hecke eigen-functions for affine Kac-Moody
- A partial generalization of the GEOMETRIC Satake isomorphism to the 
affine case
- Eisenstein series for affine Kac-Moody groups

October 25 (Thursday), 4:30 p.m, room E 206.
Alexander Braverman (Brown University).
Hecke operators for affine Kac-Moody groups

                  Abstract

The starting point for the Langlands conjecutres is the Satake isomorphism
which describes the spherical Hecke algebra of a sprit redusctive
p-adic group G in terms of the representation ring of its Langlands dual 
group. In this talk we shall generalize this result to the case when G is
an affine Kac-Moody group. Depending on how much time is left the 
following topics may also be discussed:
- Hecke operators and Hecke eigen-functions for affine Kac-Moody
- A partial generalization of the GEOMETRIC Satake isomorphism to the 
affine case
- Eisenstein series for affine Kac-Moody groups

October 29 (Thursday), 4:30 p.m, room E 206.

Vadim Vologodsky. The conjugate filtration on the ring of
differential operators in characteristic p.

                   Abstract

The ring D of differential operators in characteristic p,
besides the order filtration, has another natural filtration induced by a 
filtration on the center of D. The associated graded algebra is a 
canonically split Azumaya algebra over the cotangent space. I will explain

how this construction combined with the formalism of filtered derived 
categories leads to a generalization of Katz's formula relating the 
p-curvature and the Kodaira-Spencer operator.

This talk is based on joint work with Arthur Ogus. It will cover
essentially the same material as the talk I gave last Spring on the AG 
seminar but this time the exposition will be more detailed.

Vologodsky's talk will be on MONDAY.
Sorry for the mistake in the original announcement.

MONDAY (October 29), 4:30 p.m, room E 206.

Vadim Vologodsky. The conjugate filtration on the ring of
differential operators in characteristic p.

                   Abstract

The ring D of differential operators in characteristic p,
besides the order filtration, has another natural filtration induced by a 
filtration on the center of D. The associated graded algebra is a 
canonically split Azumaya algebra over the cotangent space. I will explain

how this construction combined with the formalism of filtered derived 
categories leads to a generalization of Katz's formula relating the 
p-curvature and the Kodaira-Spencer operator.

This talk is based on joint work with Arthur Ogus. It will cover
essentially the same material as the talk I gave last Spring on the AG 
seminar but this time the exposition will be more detailed.

No seminar on Thursday (November 1).

The remaining meetings of the seminar in autumn will be devoted to
studying Jacob Lurie's book " Higher Topos Theory",
which can be downloaded from

http://www-math.mit.edu/~lurie/papers/highertopoi.pdf

We will mostly study the first chapter,
which is devoted to higher category theory,
as developed by Boardman-Vogt and Joyal.

On Monday (November 5), Alexander Beilinson will give an introductory
lecture. The same day Mitya Boyarchenko will begin his series of
detailed lectures.

As usual, we meet in room E 206 at 4:30 p.m.

Second year students are welcome. The only prerequisite is
some familiarity with the notions of
simplicial set and geometric realization.

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

Mitya Boyarchenko will continue his series of talks on
infinity-categories.

                   Abstract

Here are some related announcements.

1. Some notes related to the first lecture are posted at

http://www.math.uchicago.edu/~mitya/langlands/quasicategories.pdf

Hopefully these notes will be continuously updated as the lecture
series progresses. At present the notes contain four sections that
recall some background from basic category theory and the theory of
simplicial sets used in the theory of infinity-categories. In
particular, the notes contain detailed explanations of all the
things that were mentioned in Mitya's lecture given on 11/05.


2. These notes also contain a list of 22 exercises (which will be
updated and expanded in the future), which are rather instructive
and may help the reader digest the material. Some of these exercises
were suggested by Jacob Lurie. The reader familiar with the current
version of the notes will be able to solve the first 13 exercises,
and we recommend doing so before the next lecture (in case you are
encountering the material explained in the notes for the first time).


3. The notes for a series of lectures on spectra and K-theory given
by Mitya Boyarchenko in the spring of 2006 can now be downloaded
from the seminar webpage (near the bottom of the page):

http://www.math.uchicago.edu/~mitya/langlands.html

These notes have a nonempty overlap with the topics that have been
and will be discussed in the seminar this quarter. In particular,
the notes contain some background on the theories of simplicial
sets and model categories.


4. The theory of infinity-categories underlies Jacob Lurie's approach
to derived algebraic geometry. Some further motivation for studying
derived algebraic geometry can be found in a video of Lurie's lecture
"Bezout's theorem and nonabelian homological algebra", available at

http://streamer.cit.utexas.edu/math-grasp/lurie.html

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

Mitya Boyarchenko will continue his series of talks on
infinity-categories.

                    Abstract

This lecture and the next one will be devoted to developing the basic
language of category theory in the setting of infinity-categories
(i.e., simplicial sets satisfying the weak Kan property).

I will begin by explaining the definition of objects and morphisms in an
infinity-category. The first fundamental difference between
infinity-categories and ordinary categories is that in the former,
compositions of morphisms are defined non-uniquely; rather, there
is a contractible space of choices for the composition.

The next "section" will be devoted to a detailed explanation of the
following picture.

An infinity-category X gives rise to ordinary or enriched categories in
several different ways. At the "most rigid" level, it gives rise to a
category C[X] enriched over simplicial sets (which can then be turned into
a topological category using the geometric realization functor). By
remembering only the homotopy types of the morphism
spaces in C[X], we obtain a category h(X) enriched over the homotopy
category of spaces. Finally, by remembering only \pi_0 of the
morphism spaces in h(X), we obtain an ordinary category, which turns out
to be isomorphic to the Poincare category of the simplicial set X.
Moreover, this category admits an alternate construction (which is only
valid when X is an infinity-category) that is more natural and transparent
than the construction mentioned in the previous talk.

If time permits, I will also explain the infinity-categorical analogues of
the notions of a commutative diagram and of an isomorphism between
objects.

Monday (November 12), 4:30 p.m, room E 206.

Mitya Boyarchenko. Infinity-categories.III.

Although Mitya's lecture will be self-contained,
I strongly recommend to gain some familiarity with the material
BEFORE the lecture. You can do this by reading
sections 1.1.5, 1.2.2 and 1.2.3 of Lurie's book "Higher Topos Theory",
available at the following URL:

http://www-math.mit.edu/~lurie/papers/highertopoi.pdf


                     Abstract


At the end of the previous lecture I gave an informal introduction to the
notion of the topological nerve of a topological category (that is, a
category enriched over topological spaces). One can prove that this nerve
is always an infinity-category (i.e., a simplicial set satisfying the weak
Kan extension property). Thus the topological nerve construction gives us
a way to pass from topological categories to infinity-categories.

It turns out that if C is a topological category, its topological nerve
depends only on the simplicial category Sing(C) obtained from C by
replacing all the Hom spaces with their total singular complexes. I will
begin my lecture with a precise definition of the simplicial nerve of a
simplicial category. Then I will explicitly describe the left adjoint to
the simplicial nerve functor. This left adjoint allows us to define the
"homotopy category" of any simplicial set.

If X is an infinity-category, the morphisms in the homotopy category of X
and in the fundamental (Poincare) category of X admit a very
simple and explicit description, which will be explained in my talk.

The overall goal of this lecture will be to understand the relationship
between the three "models" of the notion of a higher category, namely,
infinity-categories, topological categories and simplicial categories.

An updated version of Mitya Boyarchenko's notes on infinity-categories
(a.k.a. quasi-categories) has been posted online at the following URL:

http://www.math.uchicago.edu/~mitya/langlands/quasicategories.pdf

The first section of these notes is a summary of Mitya's lecture given
last Thursday. The sections containing background material have been
revised. The list of exercises has been expanded (there are now 39). All
comments, corrections and suggestions regarding the notes are welcome.

Thursday (November 15), 4:30 p.m, room E 206.

Mitya Boyarchenko. Infinity-categories. IV.


                   Abstract


In this lecture I will discuss the notions of homotopy coherent and
homotopy commutative diagrams in infinity-categories, and homotopy
(co)limits of diagrams in infinity-categories. If time permits, I
will state Yoneda's lemma for infinity-categories.

One of the goals of this lecture will be to supply the audience with
enough background for solving the exercises on infinity-categories
suggested by Jacob Lurie (they appear in the last section of the
current version of my notes, available on the seminar webpage).

There will be no more meetings of the
Langlands seminar this quarter.

However there will be an informal seminar on infinity-categories.
Here is the announcement of the first talk.

Monday (November 19), 4:30 p.m, room E 206.
Vadim Vologodsky. Introduction to stable infinity-categories.

Abstract: I will explain the notion of stable infinity-category following
Lurie's paper "Derived Algebraic Geometry 1: Stable infinity-categories".
I will also explain that the homotopy category of a stable category  has
a structure of a triangulated category and how an abelian category A gives
rise to a stable category whose homotopy category is equivalent to
the derived category D(A).

For your information:

---------- Forwarded message ----------

2008 TALBOT WORKSHOP: AFFINE LIE ALGEBRAS AND CHIRAL STRUCTURES
March 30 - April 5, 2008
Plymouth, Massachusetts

This posting announces the 2008 Talbot Workshop, Affine Lie algebras and
chiral structures, mentored by Dennis Gaitsgory.

The workshop will constitute a weeklong retreat with talks and organized 
discussions during the mornings and evenings; the afternoon schedule will
be
kept clear for informal discussions and collaborations. The general focus of
the workshop will be geometric approaches to representations of affine 
Kac-Moody Lie algebras using Beilinson and Drinfeld's theory of chiral and

factorization algebras. A particular focus will be recent work of
Gaitsgory and
Lurie on the derived geometric Satake equivalence and a quantum
formulation of
the geometric Langlands conjecture. The workshop discussions will have an 
expository character and will be aimed at advanced graduate students and
junior
faculty interested in this area.

Further information about the workshop will become available at:

http://www-math.mit.edu/~jnkf/talbot

This website also contains information about the past Talbot workshops: 
Topological modular forms (mentored by Michael Hopkins), Automorphisms of 
manifolds (mentored by Michael Weiss), Geometric Langlands (mentored by
David
Ben-Zvi), and Geometric models of elliptic cohomology (mentored by Stephan

Stolz).

If you are interested in participating or would like to receive further
information about the workshop, please email Owen Gwilliam (gwilliam (at)
math.northwestern.edu). We may be able to subsidize participants' travel
expenses, pending renewal of our NSF grant. Please note that the number of
participants will be strictly limited for space reasons.

Organizers:
John Francis (jnkf(at)math.mit.edu)
Owen Gwilliam (gwilliam(at)math.northwestern.edu)

A first draft of the notes for Mitya Boyarchenko's lectures on
infinity-categories is now available on the seminar webpage, at the 
following URL:

http://www.math.uchicago.edu/~mitya/langlands/quasicategories.pdf

The notes include everything that was covered in Mitya's talks, plus some 
background material and a long list of exercises (some of which were 
suggested by Jacob Lurie).

A comment for those who attended Mitya's third lecture (devoted to the 
simplicial nerve and simplicial Poincare category functors): in the notes,

a way of understanding the simplicial Poincare category functor (which 
assigns a simplicial category to every simplicial set) is explained which 
is different from, and, hopefully, more concise and comprehensible, than 
the one presented in Mitya's lecture.

All feedback on these notes is welcome (please send it to
mitya@math.uchicago.edu).

Kobi Kremnizer (MIT) will give 3 talks on
the analog of Beilinson-Bernstein localization for quantum groups:
  January 10 (Thursday), 4:30 p.m.,
  January 11 (Friday), 4:00 p.m.,
  January 14 (Monday), 4:30 p.m.

                  Abstract

I will give 3 talks on localization for quantum groups in
the generic case and in the case of roots of unity.

In the first talk I will review some
noncommutative algebraic geometry (noncommutative projective varieties) and
will consider the quantum flag variety as an example . The quantum flag
variety will be defined as an equivariant object and I will prove that it
is
projective. Then I will introduce the category of quantum D-modules on the
quantum flag variety and I will prove a Beilinson-Bernstein localization 
result for the quantum group. In later talks I will describe the root of 
unity case and its relation to derived algebraic geometry.

January 10 (Thursday), 4:30 p.m, room E 206.

Kobi Kremnizer (MIT). Quantum localization. I.

The second talk will be given on Friday (Jan 11) at 4:30 p.m.
(NOT at 4 p.m., as I wrote in the first announcement).

The third talk will be given on Monday at 4:30 p.m.


                  Abstract

I will give 3 talks on localization for quantum groups in
the generic case and in the case of roots of unity.

In the first talk I will review some noncommutative algebraic geometry
(noncommutative projective varieties) and will consider the quantum flag
variety as an example . The quantum flag variety will
be defined as an equivariant object and I will prove that it is
projective. Then I will introduce the category of quantum D-modules on the
quantum flag variety and I will prove a Beilinson-Bernstein localization 
result for the quantum group. In later talks I will describe the root of 
unity case and its relation to derived algebraic geometry.

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

Kobi Kremnizer (MIT). Quantum localization. II.

January 14 (Monday), 4:30 p.m, room E 206.

Kobi Kremnizer (MIT). Quantum localization. III.

No seminar on Jan 17 (Thursday) and Jan 21 (Monday).

Dennis Gaitsgory will speak on January 24 and 28.

No seminar today and on Monday (Jan 21).

Dennis Gaitsgory will speak on January 24 (i.e., NEXT Thursday)
and on January 28.

Title of his talks: Langlands duality for quantum groups.

                         Abstract.

Let G be a reductive group over a field of characteristic zero,
and let ^LG be its Langlands dual. The category of algebraic
representations of ^LG can be realized geometrically via
spherical perverse sheaves on the affine Grassmannian of G,
denoted Gr_G.

In these talks we will explain the idea of Jacob Lurie of how to
realize geometrically the category of representations of the
quantum group corresponding to ^LG. The answer will be given by
the twisted Whittaker sheaves on Gr_G, twisted by the determinant
line bundle.

The passage between the two categories will be realized through
the category of Factorizable Sheaves, introduced by Finkelberg and
Schechtman.

January 24 (Thursday), 4:30 p.m, room E 206.

Dennis Gaitsgory (Harvard).
Langlands duality for quantum groups. I.
(Dennis will give his second talk on Jan 28, Monday).

                         Abstract.

Let G be a reductive group over a field of characteristic zero,
and let ^LG be its Langlands dual. The category of algebraic
representations of ^LG can be realized geometrically via
spherical perverse sheaves on the affine Grassmannian of G,
denoted Gr_G.

In these talks we will explain the idea of Jacob Lurie of how to
realize geometrically the category of representations of the
quantum group corresponding to ^LG. The answer will be given by
the twisted Whittaker sheaves on Gr_G, twisted by the determinant
line bundle.

The passage between the two categories will be realized through
the category of Factorizable Sheaves, introduced by Finkelberg and
Schechtman.

January 28 (Monday), 4:30 p.m, room E 206.

Dennis Gaitsgory (Harvard).
Langlands duality for quantum groups. II.

No seminar on Thursday.

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

Vladimir Baranovsky (University of California, Irvine)
Algebraization of Bundles on Non-Proper Schemes.

                          Abstract

Let X be a smooth projective variety over a field k, Z a closed subset in
X and U its open complement.  Denote by A_r the ring of truncated
polynomials k[x]/x^{r+1} and by A the ring k[[x]] of  formal Taylor series
in x.

Suppose that for every nonnegative r we are given a vector bundle F_r on 
U x Spec(A_r)  such that the restriction of F_r to U x
Spec(A_{r-1}) is isomorphic to F_{r-1}.

When codimension of Z in X is at least 3, we show that this sequence
arises from a vector bundle F on an appropriate open subset of U x
Spec(A). We give an example showing that for codimension 2 the similar
statement fails and formulate additional boundedness conditions that
ensure existence of F.

We will also explain how these results relate to the construction of the
Uhlenbeck stack, a (still somewhat conjectural) compactification of the
stack of vector bundles on X.

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

Vladimir Baranovsky (University of California, Irvine)
Algebraization of Bundles on Non-Proper Schemes.

                          Abstract

Let X be a smooth projective variety over a field k, Z a closed subset in
X and U its open complement.  Denote by A_r the ring of truncated
polynomials k[x]/x^{r+1} and by A the ring k[[x]] of  formal Taylor series
in x.

Suppose that for every nonnegative r we are given a vector bundle F_r on 
U x Spec(A_r)  such that the restriction of F_r to U x
Spec(A_{r-1}) is isomorphic to F_{r-1}.

When codimension of Z in X is at least 3, we show that this sequence
arises from a vector bundle F on an appropriate open subset of U x
Spec(A). We give an example showing that for codimension 2 the similar
statement fails and formulate additional boundedness conditions that
ensure existence of F.

We will also explain how these results relate to the construction of the
Uhlenbeck stack, a (still somewhat conjectural) compactification of the
stack of vector bundles on X.

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

David Kazhdan (Hebrew University)
will give a very informal introduction to the Fundamental Lemma.

I strongly recommend to attend his talk.

Later in February there will be more technical introductory
talks on the Fundamental Lemma,
and in March Ngo Bao Chau will speak on his proof of the Lemma.

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

David Kazhdan (Hebrew University)
A very informal introduction to the Fundamental Lemma. II.

No seminar on Thursday (Feb 14).

On Monday (February 18) Tasho Kaletha (University of Chicago)
will give a talk on the formulation of the Fundamental Lemma.
It will be slower and more detailed than Kazhdan's talks.
The goal is to prepare us for Ngo's talks in March.

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

Tasho Kaletha (University of Chicago)
will speak on the formulation of the Fundamental Lemma.
It will be slower and more detailed than Kazhdan's lectures.
He will not assume knowledge of the material explained by Kazhdan.

Kaletha's Monday talk will be the first one in the series whose
goal is to prepare us for Ngo's lectures in March.

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

Tasho Kaletha (University of Chicago)
will speak on the formulation of the Fundamental Lemma
following Section 1 of Ngo Bao Chau's e-print.
You can find the e-print at the seminar web page
  http://www.math.uchicago.edu/~mitya/langlands.html
in the section "Fundamental Lemma and Endoscopy".
There are also other materials at the web page that may
help understand the talks by Kazhdan and Kaletha:

my notes on the local Tate-Nakayama duality
and the article by Kottwitz to which Ngo refers;

my notes on what Ngo calls "Kostant's section"
and the related preprint by Beilinson and me;

my notes on on centralizers of semisimple elements of
semisimple groups and their relation to extended Dynkin diagrams.

More materials of this type will probably appear at the seminar web page
in the future.

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

Tasho Kaletha will give his second talk
on the formulation of the Fundamental Lemma.
He will follow Section 1 of Ngo Bao Chau's 2008 e-print.

1. On February 25 (Monday) I will give a talk whose goal is
to explain Section 2 of
Ngo Bao Chau's article on the Fundamental lemma
and maybe some other parts of the article (e.g.,
Subsections 3.3, 4.3)

As usual, we meet at 4:30 p.m in room E 206.

2. At the seminar web page
 http://www.math.uchicago.edu/~mitya/langlands.html
in the section "Fundamental Lemma and Endoscopy"
you can find one more file written by me.
I reformulate there (a part of) the fundamental Lemma
for SL(n) in very concrete and "classical" terms.
This formulation goes back to the article
R. Kottwitz, Unstable orbital integrals on SL(3),
Duke Math. J. 48 (1981), no. 3, 649-664.

On Thursday  (February 28) I will explain
some parts of Section 4 of Ngo Bao Chau's work.
This section is about the Hitchin fibration.

As usual, we meet at 4:30 p.m in room E 206.

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

Ngo Bao Chau (University Paris-Sud, Institute for Advanced Study).
Hitchin fibration and fundamental lemma. I.

                   Abstract

In my talks, I will explain the elements of the proof of
Langlands-Shelstad's
fundamental lemma for Lie algebra based on the geometry of the
Hitchin fibration. The basic ingredient is the description of
the supports of the simple perverse sheaves which occur in the
decomposition of the l-adic cohomology of the Hitchin  fibration.

1. Today Ngo Bao Chau will give his first talk on the proof of the
fundamental lemma (4:30 p.m, room E 206).

2. On the seminar web page you can find some new materials:

(i) a slightly revised vesrion of my notes on the fundamental lemma for
SL(n),

(ii) the notes of my lecture on regular centralizers
(Section 2 of Ngo's article),

(iii) the notes of my lecture on the Hitchin fibration
(a part of Section 4 of Ngo's article).

Thursday (March 6), 4:30 p.m, room E 206.

Ngo Bao Chau (University Paris-Sud, Institute for Advanced Study). 
Hitchin fibration and fundamental lemma. II.

No more seminar this quarter.

This week Prof. Claude Sabbah (Ecole Polytechnique) will give informal
talks on wild twistor D-modules. The first informal talk will be given
TOMORROW (Tuesday, March 25) at 11:30 in room E206. The times of the next
informal talk(s) will be announced tomorrow.

Next week Prof. Sabbah will give lectures on twistor D-modules at the 
Geometric Langlands seminar. They will be given on
Monday (March 31) and Thursday (April 3) at 4:30 p.m. (room E206). They
will be independent of the informal talks of the first week, and will
mainly treat the case of regular (i.e. not wild, or irregular) twistor
D-modules.

The titles and abstracts of the informal talks and seminar talks are below.

                   **************************

    Informal talks (this week, starting on Tuesday,11:30 p.m., room E206)
         Title: Towards wild twistor D-modules

                           Abstract

The talks will try to motivate the need for wild objects in the theory of
twistor D-modules.
Starting from the example of a variation of complex Hodge structure, its
behaviour under Fourier-Laplace transform already exhibits many of the
features of the wild theory. I will start from the first tentative by
Deligne (1984) to introduce a Hodge filtration for irregular
singularities, and I will end by giving some ideas on the very recent work
of T. Mochizuki which eventually solves a conjecture of Kashiwara, among
others.
General definitions will be introduced on purpose, but would be
developed next week, when global motivations for the theory
(like various extensions of the Hard Lefschetz theorem) will be
discussed. These informal talks would mainly give more
motivation to the talks of the second week.

                   **************************

Talks at the Geometric Langlands seminar
on Monday (March 31) and Thursday (April 3) at 4:30 p.m. (room E206).

Title: Introduction to twistor D-modules.

                           Abstract

Starting from the classical Hard Lefschetz theorem, I will review
past work of Simpson, which justifies the notion of variation of
polarized twistor structure. Twistor D-modules are a singular
extension of such objects, and their role in the complete analytic proof, 
by T. Mochizuki, of a conjecture of Kashiwara extending the
decomposition theorem of Beilinson-Bernstein-Deligne-Gabber to
semisimple perverse sheaves will be explained. The talks are
independent of the informal talks of the first week, and will
mainly treat the case of regular (i.e. not wild, or irregular)
twistor D-modules.

Monday (March 31), 4:30 p.m, room E 206.

Claude Sabbah (Ecole Polytechnique)
Introduction to twistor D-modules. I.

            Abstract

Starting from the classical Hard Lefschetz theorem, I will review
past work of Simpson, which justifies the notion of variation of
polarized twistor structure. Twistor D-modules are a singular
extension of such objects, and their role in the complete analytic proof, 
by T. Mochizuki, of a conjecture of Kashiwara extending the
decomposition theorem of Beilinson-Bernstein-Deligne-Gabber to
semisimple perverse sheaves will be explained. The talks are
independent of the informal talks of the previous week, and will
mainly treat the case of regular (i.e. not wild, or irregular)
twistor D-modules.

Monday (March 31), 4:30 p.m, room E 206.

Claude Sabbah (Ecole Polytechnique)
Introduction to twistor D-modules. I.

            Abstract

Starting from the classical Hard Lefschetz theorem, I will review
past work of Simpson, which justifies the notion of variation of
polarized twistor structure. Twistor D-modules are a singular
extension of such objects, and their role in the complete analytic proof,
by T. Mochizuki, of a conjecture of Kashiwara extending the
decomposition theorem of Beilinson-Bernstein-Deligne-Gabber to
semisimple perverse sheaves will be explained. The talks are
independent of the informal talks of the previous week, and will
mainly treat the case of regular (i.e. not wild, or irregular)
twistor D-modules.

1. Claude Sabbah will give his second talk on twistor D-modules
on Thursday (April 3), 4:30 p.m, room E 206.

2. Attached is a PDF file with my notes,
in which I try to explain to myself the notion of
variation of twistor structures from Sabbah's Monday lecture.

3. On Monday I mentioned Russian works in mathematical physics, in
which the nonlinear Laplace equation for a map from a Riemann surface
to a Lie group (or to a symmetric space) was interpreted as a zero
curvature condition. Here are the references:

Zakharov, V. E.; Mikhailov, A. V.
Relativistically invariant two-dimensional models of field theory which
are integrable by means of the inverse scattering problem method. Soviet
Phys. JETP 74 (1978), no. 6, 1953--1973

Zaharov, V. E.; Shabat, A. B.
Integration of the nonlinear equations of mathematical physics by the
method of the inverse scattering problem. II.  Funktsional. Anal. i
Prilozhen. 13 (1979), no. 3, 13--22.

In the latter work the nonlinear Laplace equation appears as
equation (2.10).

Attachment: Twistor structures.pdf
Description: Adobe PDF document

No seminar next week.

Beilinson will begin his series of talks on April 14 (Monday).
Title of this series:
Epsilon-factors for the period determinants on curves.

             Abstract

Long ago, Deligne mentioned that irregularity of a connection
is analogous to wild ramification in arithmetic situation. Pursuing the 
idea, he suggested that the derminants of period matrices, similarly to 
the constants in the functional equations of L-series, should admit a 
factorization that would manifest their hidden local nature. This program 
was completed in a work of Bloch, Esnault, and Deligne (2005), which used 
a version of the stationary phase method

I will describe a different approach. Namely, the epsilon-factors  are
defined directly using Fredholm determinants and  certain ideas  from
class field theory. The product formula is proved by a global  argument
(based on a result of Goldman and Pickrell-Xia about
ergodicity of the action of the Teichmuller group on the moduli of 
unitary local systems).

Monday (April 14), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the period determinants on curves. I.

             Abstract

Long ago, Deligne mentioned that irregularity of a connection
is analogous to wild ramification in arithmetic situation. Pursuing the 
idea, he suggested that the derminants of period matrices, similarly to 
the constants in the functional equations of L-series, should admit a 
factorization that would manifest their hidden local nature. This program 
was completed in a work of Bloch, Esnault, and Deligne (2005), which used 
a version of the stationary phase method

I will describe a different approach. Namely, the epsilon-factors  are
defined directly using Fredholm determinants and  certain ideas  from
class field theory. The product formula is proved by a global  argument
(based on a result of Goldman and Pickrell-Xia about
ergodicity of the action of the Teichmuller group on the moduli of 
unitary local systems).

Thursday (April 17), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. II.

Monday (April 21), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. III.

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

Monday (April 28), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. IV.

Monday (April 28), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. IV.

Claude Sabbah will give
his second informal talk on wild twistor D-modules
tomorrow (Wednesday), 11:30 p.m., room E206

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

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. V.

Monday (May 5), 4:30 p.m, room E 206.

Alexander Beilinson.
Epsilon-factors for the determinants of periods on curves. VI.

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

2. Tomorrow (Tuesday, May 6) at 4:30 p.m in room E203
Jacob Lurie (MIT) will give a talk at the topology seminar on
"Algebraic Groups over the Sphere Spectrum"
(based on joint work with Dennis Gaitsgory).

Abstract of Lurie's talk:
Let G be a compact Lie group. Then the complexification of G has the
structure of a reductive algebraic group over the field C of complex
numbers. This algebraic group is canonically defined over the ring of
integers Z. In this talk, I will discuss the problem of defining this
group "over the sphere spectrum".

3. A conference in honour of R.P. Langlands
"Current developments and directions in the Langlands program"
will be held at Northwestern University on May 10-14.
You can find the conference schedule and other information at
http://www.math.northwestern.edu/~emerton/Langlands.html