This is an archive of email messages concerning the Geometric Langlands Seminar for 08-09.

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.).

First meeting: Thursday, October 2.

On October 2 I will give an introductory talk on braided "fusion
categories"
(i.e., braided tensor categories satisfying semisimplicity and finiteness
conditions).

Then I will give an overview of the theory of braided fusion categories.
The main questions are how one can think of them and how one can try to
classify all of them.

There will also be a talk by Kobi Kremnizer on the modular categories
coming from quantum groups at roots of unity. (Modular categories are,
roughly speaking, those braided fusion categories which are as far as
possible
from being symmetric).

On the other hand, Mitya Boyarchenko will speak on the theory of character

sheaves on unipotent groups. These sheaves are grouped into finite "blocks",
and character sheaves from a given block are, in fact, irreducible objects

of a modular category (this is one of the reasons why I got interested in 
the theory of modular categories and, more generally, braided fusion
categories).

Later (maybe in winter) there will be talks about the relation between 
modular categories and topology in dimensions 2 and 3.

Of course, there will also be talks on other subjects.

Thursday (Oct 2), 4:30 p.m, room E 206.

I will give a very elementary introductory talk whose
goal is to recall the definition
of monoidal category, braided category, rigidity, etc.

Literature:
the beginning of J. Bernstein's Sacler lectures:
   http://front.math.ucdavis.edu/9501.5172
and the beginning of Lectures on tensor categories,
    by D.Calaque and P.Etingof:
http://front.math.ucdavis.edu/0401.5246

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

In the next talks I will give an overview of the theory of braided fusion
categories, i.e., braided tensor categories satisfying semisimplicity and
finiteness conditions. Because of these conditions, a braided fusion
category is a very "concrete" thing (basically, a collection of matrices 
satisfying certain identities). In my talks  I will try to explain how one
can try to reduce classifying all such categories to classifying "simple" 
ones (the notion of simple braided fusion category is parallel to the
notion of simple finite group; in particular, "simple" is a synonym for
"mysterious" and "difficult"). As a heuristic tool, I will use an analogy 
between braided fusion categories and Casimir Lie algebras
(i.e., pairs consisting of a Lie algebra g and a g-invariant element of
the symmetric square of g).

I have to cancel my Thursday talk
to avoid conflict with the lecture "On Mean Field Games"
by Pierre-Louis Lions on Thursday, October 2 at 4:30 PM
(this is the first of his Amick Lectures).
I will give my first talk on Monday, October 6 at 4:30 p.m.

I am very sorry for the confusion
(unfortunately, I didn't notice before that the first lecture
by Lions is on Thursday).

> I will give a very elementary introductory talk whose
> goal is to recall the definition
> of monoidal category, braided category, rigidity, etc.
>
> Literature:
> the beginning of J. Bernstein's Sacler lectures:
>    http://front.math.ucdavis.edu/9501.5172
> and the beginning of Lectures on tensor categories,
>     by D.Calaque and P.Etingof:
> http://front.math.ucdavis.edu/0401.5246
>
>      ************
>
> In the next talks I will give an overview of the theory of braided
fusion
> categories, i.e., braided tensor categories satisfying semisimplicity
and
> finiteness conditions. Because of these conditions, a braided fusion
category is a very "concrete" thing (basically, a collection of matrices
satisfying certain identities). In my talks  I will try to explain how
one
> can try to reduce classifying all such categories to classifying
"simple"
> ones (the notion of simple braided fusion category is parallel to the
notion of simple finite group; in particular, "simple" is a synonym for
"mysterious" and "difficult"). As a heuristic tool, I will use an
analogy
> between braided fusion categories and Casimir Lie algebras
> (i.e., pairs consisting of a Lie algebra g and a g-invariant element of
the symmetric square of g).
>
>
>
>
>
>

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

I will give a very elementary introductory talk whose
goal is to recall the definition
of monoidal category, braided category, rigidity, etc.
(this is the talk which was supposed to be last Thursday.)

Literature:
the beginning of J. Bernstein's Sacler lectures:
   http://front.math.ucdavis.edu/9501.5172
and the beginning of Lectures on tensor categories,
    by D.Calaque and P.Etingof:
http://front.math.ucdavis.edu/0401.5246

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

In the next talks I will give an overview of the theory of braided fusion
categories, i.e., braided tensor categories satisfying semisimplicity and
finiteness conditions. Because of these conditions, a braided fusion
category is a very "concrete" thing (basically, a collection of matrices 
satisfying certain identities). In my talks  I will try to explain how one
can try to reduce classifying all such categories to classifying "simple" 
ones (the notion of simple braided fusion category is parallel to the
notion of simple finite group; in particular, "simple" is a synonym for
"mysterious" and "difficult"). As a heuristic tool, I will use an analogy 
between braided fusion categories and Casimir Lie algebras
(i.e., pairs consisting of a Lie algebra g and a g-invariant element of
the symmetric square of g).

1. No seminar on Thursday (Oct. 9).


2. The composition of monoidal functors is STRICTLY associative,
see the attached PDF file written by Rina Anno.


3. I will continue on Monday (Oct 13).
First, I will briefly recall the notion of rigid monoidal categories.
Those who never encountered it are recommended to read
p. 7-15 of Mitya's notes of his first 2007 talk on modular categories and
solve at least some of the exercises therein, see
   http://www.math.uchicago.edu/~mitya/modular/modular1.pdf

Then I will tell some fairy tales about fusion categories
(i.e., k-linear rigid monoidal categories satisfying the strongest
semisimplicity and finiteness conditions).
A fusion category is a very "concrete" object (a certain kind of
linear algebra datum). I will formulate some quite interesting
general theorems on them, mostly from the following work
by Etingof, Nikshych, and Ostrik:
    http://arxiv.org/abs/math/0203060

Attachment: strict_associativity.pdf
Description: Adobe PDF document

Monday (Oct 13), 4:30 p.m, room E 206.

I will continue my talk.

First, I will briefly recall the notion of rigid monoidal categories.
Those who never encountered it are recommended to read
p. 7-15 of Mitya's notes of his first 2007 talk on modular categories and
solve at least some of the exercises therein, see
 http://www.math.uchicago.edu/~mitya/modular/modular1.pdf

Then I will switch from "abstract nonsense" to a much more
interesting and "concrete" subject.
Namely, I will tell some fairy tales about fusion categories
(i.e., k-linear rigid monoidal categories satisfying the strongest
semisimplicity and finiteness conditions).
A fusion category is a very "concrete" object (a certain kind of linear
algebra datum). I will formulate some quite interesting
general theorems on them, mostly from the following work
by Etingof, Nikshych, and Ostrik:
http://arxiv.org/abs/math/0203060

In particular:
1. Up to equavalence, there are only finitely many fusion categories C
with fixed Grothedieck ring K_0(C).

Corollary: each fusion category is defined over a number field.

2. After tensoring by the field of rational numbers Q,
the center of K_0(C) becomes a product of fields each of which
is a finite ABELIAN extension of Q.

3. For any homorphism f from K_0(C) to the field of algebraic numbers, the
ideal of the ring of algebraic integers generated by
the sum of f(X)f(X^*) (where X runs through the set of irreducible
representations) has the form au, where
a is an m-th root of an integer (for some natural number m)
and u is a unit (i.e., an algberaic integer whose inverse is also
an algberaic integer).

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

I will discuss two notions of dimension in a fusion category C.
One of them (called quantum or categorical dimension) is
a homomorphism from K_0 (C) to k; it depends on the choice of
an additional structure on C. The other one is a canonical homomorphism
from K_0 (C) to the real numbers.

In particular, I will formulate deep results by Etingof, Nikshych, and
Ostrik on the number-theoretic properties of the dimensions.
For instance, they proved that the dimensions belong to the maximal
abelian extension of Q.

I will also discuss duality in fusion categories, in particular,
Radford's isomorphism between X and X^{****} (the fourth dual of X).

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

I will give my last talk on fusion categories.

In this talk I will formulate the results by Etingof, Nikshych, and Ostrik
on the number-theoretic properties of the dimensions
and discuss Radford's isomorphism between
X and X^{****} (the fourth dual of X).

If time permits I will discuss the proofs of the positivity theorems
formulated on Thursday.

No seminar on Thursday.
Kobi Kremnizer will speak on Monday (October 27).

Attached are two PDF files related to my last talk:

(i) a proof of the Frobenius-Perron theorem,
(ii) a proof of the following property of
fusion categories: for every object X some power of X contains
the unit object. The proof of this property (due to Etingof-Nikshych)
uses the *-algebra technique. Please tell me if you know a more direct proof.

Attachment: Frobenius-Perron.pdf
Description: Adobe PDF document

Attachment: Etingof-Nikshych.pdf
Description: Adobe PDF document

Monday (Oct 27), 4:30 p.m, room E 206.

Kobi Kremnizer. Tannakian categories

             Abstract

In this talk I will review the possible notions of symmetry on a
monoidal category. Then I will talk about fiber functors and the
definition of a Tannakian category and I will give examples. I will
explain Deligne's theorem on Tannakian duality and his theorem on a
necessary and sufficient condition for the existence of a fiber functor in
terms of dimensions of objects. In order to indicate the proof of the
first theorem I will recall the Barr-Beck formalism. For the second
theorem I will explain the notion of algebraic geometry relative to a
symmetric monoidal category.

No seminar on Thursday.

Kobi Kremnizer will continue next Monday, Nov 3.

The articles on Tannakian categories are available online.

The link for the Deligne-Milne paper is
http://www.jmilne.org/math/Books/DMOS.pdf

The links for Deligne's paper are
http://www.math.uchicago.edu/~drinfeld/Deligne.pdf
http://www.springerlink.com.proxy.uchicago.edu/content/nh670v/

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

Kobi Kremnizer will continue and probably finish his talk on
Tannakian categories.



The articles on Tannakian categories are available online.

The link for the Deligne-Milne paper is
http://www.jmilne.org/math/Books/DMOS.pdf

The links for Deligne's paper are
http://www.math.uchicago.edu/~drinfeld/Deligne.pdf
http://www.springerlink.com.proxy.uchicago.edu/content/nh670v/

No seminar on Thursday (Nov 6).

Pavel Etingof (MIT) will give a colloquium talk
this Friday at 4pm (room E206)
and he will speak at our seminar on Monday, November 10.

Below is the title and abstract of his colloquium talk.
I will send you the title and abstract of his seminar talk later.


Title of Etingof's colloquium talk: Orbifold Hecke Algebras

                 Abstract

To a group G acting discretely on a simply connected complex  manifold X,
I will attach a Hecke algebra H_q(G,X), which is a deformation
of the group algebra of G. We will see that if H^2(X,C)=0 then this 
deformation is flat. We will also see that this setting unifies many known
types of Hecke algebras - usual (finite), affine, double affine
(Cherednik), Hecke algebras of complex reflection groups
(Broue-Malle-Rouquier), and many others. In particular, there are orbifold
Hecke algebras which provide quantization of Del Pezzo surfaces and their 
Hilbert schemes.

Monday (Nov 10), 4:30 p.m, room E 206.


Pavel Etingof (MIT). Solvable fusion categories and a categorical analog of
Burnside's theorem.

                 Abstract

The goal of this talk is to explain the classical
representation-theoretic proof of Burnside's theorem in finite group 
theory, stating that a finite group of order $p^aq^b$ (where $p,q$ are
primes) is solvable, and then define the notion of a solvable
fusion category and explain how to generalize Burnside's theorem
to the categorical setting. This generalization is joint work
with D. Nikshych and V. Ostrik, and is given in the paper math/0809.3031

Pavel Etingof wrote the notes of the talk that he is going to give on
Monday. You can download the file from the seminar web page
   http://www.math.uchicago.edu/~mitya/langlands.html
It is in a new subsection titled "Fusion Categories"
(it is the fourth subsection, counting from the bottom
of the page).


> Pavel Etingof (MIT). Solvable fusion categories and a categorical analog
> of Burnside's theorem.
>
>
>                  Abstract
>
> The goal of this talk is to explain the classical
> representation-theoretic proof of Burnside's theorem in finite group
> theory, stating that a finite group of order $p^aq^b$ (where $p,q$ are
> primes) is solvable, and then define the notion of a solvable
> fusion category and explain how to generalize Burnside's theorem
> to the categorical setting. This generalization is joint work
> with D. Nikshych and V. Ostrik, and is given in the paper math/0809.3031
>
>
>
>

No seminar on Thursday (November 13).

On Monday (November 17) Kobi Kremnizer will speak on the modular
categories corresponding to quantum groups.

Monday (Nov 17), 4:30 p.m, room E 206.

Kobi Kremnizer. Modular tensor categories arising from
quantum groups at roots of unity.


                             Abstract
In this talk I will explain how to construct a modular tensor
category from the category of representations of a quantum group
at a root of unity. I will start by explaining what are quantum groups 
giving sl_2 as an example. Then we will look at the category of modules
when the parameter is a root of unity. The modular category will be a
certain subquotient of the root of unity category. To understand the
construction I will introduce the notion of  tilting modules and study
their proporties.
This story has an analogue for reductive groups in positive
characteristic.
I will remark on that as well.

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

Kobi Kremnizer will continue his talk on
modular tensor categories arising from quantum groups at roots of unity.


>                              Abstract
> In this talk I will explain how to construct a modular tensor
> category from the category of representations of a quantum group
> at a root of unity. I will start by explaining what are quantum groups
> giving sl_2 as an example. Then we will look at the category of modules
> when the parameter is a root of unity. The modular category will be a
> certain subquotient of the root of unity category. To understand the
> construction I will introduce the notion of  tilting modules and study
> their proporties.
> This story has an analogue for reductive groups in positive
> characteristic.
> I will remark on that as well.
>
>
>

Monday (Nov 24), 4:30 p.m, room E 206.

Kobi Kremnizer will continue his talk on
modular tensor categories arising from quantum groups at roots of unity.


>>                              Abstract
>> In this talk I will explain how to construct a modular tensor
>> category from the category of representations of a quantum group
>> at a root of unity. I will start by explaining what are quantum groups
>> giving sl_2 as an example. Then we will look at the category of modules
>> when the parameter is a root of unity. The modular category will be a
>> certain subquotient of the root of unity category. To understand the
>> construction I will introduce the notion of  tilting modules and study
>> their proporties.
>> This story has an analogue for reductive groups in positive
>> characteristic.
>> I will remark on that as well.
>>
>>
>>
>

No more seminars this quarter.

Happy Thanksgiving!

This quarter begins with a series of lectures by Beilinson
(see the title and abstract below). They will start on the
SECOND week of the quarter, on Jan 12 (Monday).

Note that in February (Feb 14-15) Turaev will speak on a related subject, 
namely, 3-dimensional Topological Quantum Field Theories.

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

Title of Beilinson's series:
Modular categories and the Wess-Zumino-Witten model.

                             Abstract

In these lectures I will explain the notion of modular category and its
topological interpretation as (twisted) representations of the
Teichmuller groupoid tower. As the main example, I will consider the
Wess-Zumino-Witten (WZW) model, whose rigorous algebro-geometric
treatment is due to Tsuchiya-Ueno-Yamada (1992).
The corresponding modular category is, in fact, equivalent to the modular
category that comes from  a quantum group at a root of unity.
However, my talks will be independent of Kobi Kremnizer's
lectures in the Fall.

The first talk will mostly be devoted to basic topological framework, 
illustrated by the example of modular structure
on the category of conjugation-equivariant sheaves on a finite group G 
(this is a toy version of the WZW model).

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


A.Beilinson. Modular categories and the Wess-Zumino-Witten model. I.

                             Abstract

In these lectures I will explain the notion of modular category and its
topological interpretation as (twisted) representations of the
Teichmuller groupoid tower. As the main example, I will consider the
Wess-Zumino-Witten (WZW) model, whose rigorous algebro-geometric
treatment is due to Tsuchiya-Ueno-Yamada (1992).
The corresponding modular category is, in fact, equivalent to the modular
category that comes from  a quantum group at a root of unity.
However, my talks will be independent of Kobi's lectures in the Fall.

The first talk will mostly be devoted to basic topological framework, 
illustrated by the example of modular structure
on the category of conjugation-equivariant sheaves on a finite group G
(this is a toy version of the WZW model).

Today Beilinson begins his series of talks:

> Monday (January 12) , 4:30 p.m, room E 206.
>
>
> A.Beilinson. Modular categories and the Wess-Zumino-Witten model. I.
>
>                              Abstract
>
> In these lectures I will explain the notion of modular category and its
> topological interpretation as (twisted) representations of the
> Teichmuller groupoid tower. As the main example, I will consider the
> Wess-Zumino-Witten (WZW) model, whose rigorous algebro-geometric
> treatment is due to Tsuchiya-Ueno-Yamada (1992).
> The corresponding modular category is, in fact, equivalent to the modular
> category that comes from  a quantum group at a root of unity.
> However, my talks will be independent of Kobi's lectures in the Fall.
>
> The first talk will mostly be devoted to basic topological framework,
> illustrated by the example of modular structure
> on the category of conjugation-equivariant sheaves on a finite group G
> (this is a toy version of the WZW model).
>
>
>
>

No seminar on Thursday. Beilinson will continue next Monday.

He is writing up the notes of his talk, which will be e-mailed to you in a
few days. Meanwhile you could read the notes
  http://www.math.uchicago.edu/~mitya/modular/modular4.pdf
in which the definition of tower of groupoids only slightly differs
from the one given by Sasha yesterday.

Contrary to the previous announcement,
there will be no seminar this Monday.

Beilinson will continue next Thursday.

Thursday (January 22) , 4:30 p.m, room E 206.
Sasha Beilinson will give his second talk on
    Modular categories and the Wess-Zumino-Witten model.
His write-up of the first talk is attached.

Attachment: Modular-Sasha.pdf
Description: Adobe PDF document

Monday (January 26) , 4:30 p.m, room E 206.

Sasha Beilinson will give his first talk on the Wess-Zumino-Witten model.

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

Alexander Kuznetsov (Steklov Institute and Moscow Independent University)

      Categorical resolutions of singularities

                      Abstract

I will give a definition of a categorical resolution
of singularities and explain how such resolutions can be constructed. Also
I will discuss several versions of the notion of crepancy
in this context.

Monday (February 2) , 4:30 p.m, room E 206.

Sasha Beilinson will give his second talk on the Wess-Zumino-Witten model.

No seminar on Thursday (February 5).

Monday (February 9) , 4:30 p.m, room E 206.

Rina Anno will speak on the relation between braided categories
and some objects of 3-dimensional topology (namely, knots and tangles).
Her abstract is below.

Her talk can be considered as an introduction to Turaev's lectures on
Thursday (Feb 12, 4:30 p.m.) and Friday (Feb 13, probably at 4:00 p.m.).
(Turaev will explain how a 3-dimensional topological
quantum field theory is constructed from a modular category or
from a spherical fusion category.)


Rina Anno. Categories of tangles.

                   Abstract

I will review the definitions of braided and ribbon categories.
Then I will construct the category of framed oriented tangles labeled with
elements of a given set $S$ as a category with a certain universal
property.
This universality allows us to construct braided functors from
the category of tangles to an arbitrary ribbon category, C, by
specifying the image of each generator. Each choice of C gives an isotopy
invariant of links in $R^3$. In particular, if C is the category of modules
over the quantum group $U_q(sl_2)-mod$ then one gets the famous
Jones polynomial.

Please notice the unusual time of Turaev's second lecture!


Thursday (February 12) , 4:30 p.m, room E 206
and Friday (February 13) , 4:00 p.m, room E 206

Vladimir Turaev (Indiana University). From tensor  categories to
3-dimensional Topological Quantum Field Theories.

                         Abstract

I will explain two basic constructions of 3-dimensional Topological
Quantum Field Theories from tensor categories. One construction
starts from modular categories and uses the surgery picture of
3-manifolds. The second construction starts from appropriate spherical
tensor categories and uses triangulations of manifolds.
I will also discuss known and conjectural relations between these two
approaches.

1. This quarter there will be no more meetings of the Langlands seminar.

2. On Mondays and/or Thursdays room E206 will be used for the
Vologodsky-Kremnizer seminar. As far as I know, the next meeting
of their seminar will be on Thursday, February 19.

3. The Vologodsky-Kremnizer seminar is devoted to studying
Jacob Lurie's article on topological field theories, which can be
downloaded from
  http://www-math.mit.edu/~lurie/

This article is closely related to Turaev's talks.
More precisely, according to Lurie's message below,
Turaev's second construction of a 3-dimensional TQFT
(which starts from a sphercial tensor category)
can probably be interpreted via Theorem 1.2.16 from Lurie's article  (the
Baez-Dolan cobordism hypothesis).

Subject:   	Re: your interpretation of Turaev ?
From:   	"Jacob Lurie" <jacoblurie@gmail.com>
Date:   	Fri, February 13, 2009 12:43 pm
To:   	drinfeld@math.uchicago.edu

Dear Volodya,
I don't think that my work has anything to say about the construction via
modular tensor categories (it involves reconstructing a theory defined on
manifolds of dimension 1-2-3 from its values on a circle in the case of a
very constrained target 2-category, while the cobordism hypothesis
reconstructs a field theory in dimensions 0-1-2-3 from its values on a
point
in the case of a general target category).

As for the second construction, I can imagine that it is closely related.
You can define a 3-category of monoidal categories, bimodule categories,
functors between bimodule categories, and natural transformations. Every
monoidal category is fully dualizable in the 1-dimensional sense (its
dual is the same category with the opposite monoidal structure) and is fully
dualizable in the 2-dimensional sense if it is rigid. There is an action of
SO(2) on the collection of such 2-fully dualizable monoidal categories and
if I understand correctly the SO(2) fixed points are what people call
spherical tensor categories. The cobordism hypothesis would then tell you
that any such category determines a field theory defined oriented 0-1-2
manifolds (a closed 2-manifold will get assigned a vector space). With
some further finiteness conditions, you could extend this to a theory
which assigns numbers to 3-manifolds equipped with a nonvanishing
vector field, and with some further equivariance you can remove the
dependence on the vector field. I haven't thought about what these
finiteness or equivariance conditions mean in more concrete terms.

Cheers,

Jacob

0. Please note Namboodiri Lectures by Mladen Bestvina
on March 31, April 1, and April 2. For more details, see
http://www.math.uchicago.edu/research/abstracts/namboodiri_abstracts.shtml

1. No meetings of the Langlands seminar until April 16.
(On some Mondays and Thursdays room E206 will be used for the
Vologodsky-Kremnizer seminar on TQFT.)

2. On April 16 Lev Rozansky will explain his joint work with Kapustin
about constructing a 2-category associated with holomorphic symplectic
manifolds.

3. On April 23 Matthew Emerton (NWU) will begin his series of talks on the
p-adic Langlands program.

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

Lev Rozansky (University of North Carolina)

     A 2-category of a holomorphic symplectic manifold

                  Abstract

This is a report on the ongoing joint work with A. Kapustin. To a
holomorphic symplectic manifold X we associate a 2-category whose simplest
objects are holomorphic lagrangian submanifolds. The morphisms between two
lagrangian submanifolds having a clean intersection form, roughly
speaking,
an A-infinity deformed category of coherent sheaves on their intersection. I
will provide a detailed description of the 2-category in the case when X is
a cotangent bundle of a complex manifold U. In this case the morphisms
between two lagrangian submanifolds form the category of matrix
factorizations of the difference of their generating functions and the
resulting 2-category seems to be closely related to the category of
2-modules over the 2-ring (i.e., monoidal category) of coherent sheaves on
U
(in the spirit of derived algebraic geometry). I will also explain how the
deformation of the holomorphic symplectic structure of T*U leads to
A-infinity deformations of matrix factorization categories.

No seminar tomorrow (Monday).

On Thursday (April 23) Matthew Emerton (NWU)
will begin his series of talks
"An introduction to the p-adic Langlands program".

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

Matthew Emerton (Northwestern University)
An introduction to the p-adic Langlands program.


                       Abstract

In this talk I will describe some of the ideas behind
the p-adic Langlands program.  The p-adic Langlands program is
a fairly recent development in number theory and representation
theory, which draws its inspiration from (among other sources)
the theories of p-adic and mod p modular (or more generally, automorphic) 
forms, the theories of mod p and p-adic Galois representations (including 
the deformation theory of such Galois representations), and p-adic Hodge 
theory, as well as the classical Langlands program relating
automorphic forms to Galois representations and motives.

Until recent years, the p-adic aspects of the theory of modular
forms, Galois representations, and so on did not make especially
close contact with the Langlands program.  Indeed, the representation
theory that is so intimately bound up with the Langlands program
(primarily, the smooth representation theory of p-adic reductive
groups) did not seem like an adequate tool for describing the complexities
arising in the p-adic theory of automorphic forms and Galois
representations, or in p-adic Hodge theory (which among other things aims
at describing p-adic Galois representations of Galois groups of p-adic
fields,
representations which are outside the scope of the classical local
Langlands conjecture).

The situation is now quite different.  A number of new tools and
view-points, in local representation theory, in the theory of automorphic
forms, and in the theory of Galois representations, have been discovered,
which have allowed us to begin a kind of unification of the p-adic world
of number theory with the world of the Langlands program.  Furthermore, it
now seems reasonable to hope that in the future we will have a truly
p-adic Langlands program, which exhibits a reciprocity between all p-adic
representations of Galois groups of global fields, and systems of Hecke
eigenvalues that appear in certain p-adic spaces that serve as a genuinely
p-adic analogue of the classical spaces of automorphic forms.

In this talk, I will begin to explain some of these ideas, beginning with
an overview of what is known and what the goals are, as well as some
background history.   I will then introduce the so-called p-adically
completed cohomology spaces attached to reductive groups over number
fields, which will serve as p-adic analogues of classical spaces of
modular forms.  If time permits, I will explain the expected relationship
between these spaces and spaces of Galois representations.

No Langlands seminar on Monday.
Emerton will continue on April 30 (Thursday).

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

Matthew Emerton (Northwestern University)
An introduction to the p-adic Langlands program. II.


>                        Abstract
>
> In this talk I will describe some of the ideas behind
> the p-adic Langlands program.  The p-adic Langlands program is
> a fairly recent development in number theory and representation
> theory, which draws its inspiration from (among other sources)
> the theories of p-adic and mod p modular (or more generally, automorphic)
> forms, the theories of mod p and p-adic Galois representations (including
> the deformation theory of such Galois representations), and p-adic Hodge
> theory, as well as the classical Langlands program relating
> automorphic forms to Galois representations and motives.
>
> Until recent years, the p-adic aspects of the theory of modular
> forms, Galois representations, and so on did not make especially
> close contact with the Langlands program.  Indeed, the representation
> theory that is so intimately bound up with the Langlands program
> (primarily, the smooth representation theory of p-adic reductive
> groups) did not seem like an adequate tool for describing the complexities
> arising in the p-adic theory of automorphic forms and Galois
> representations, or in p-adic Hodge theory (which among other things aims
> at describing p-adic Galois representations of Galois groups of p-adic
> fields,
> representations which are outside the scope of the classical local
> Langlands conjecture).
>
> The situation is now quite different.  A number of new tools and
> view-points, in local representation theory, in the theory of automorphic
> forms, and in the theory of Galois representations, have been discovered,
> which have allowed us to begin a kind of unification of the p-adic world
> of number theory with the world of the Langlands program.  Furthermore, it
> now seems reasonable to hope that in the future we will have a truly
> p-adic Langlands program, which exhibits a reciprocity between all p-adic
> representations of Galois groups of global fields, and systems of Hecke
> eigenvalues that appear in certain p-adic spaces that serve as a genuinely
> p-adic analogue of the classical spaces of automorphic forms.
>
> In this talk, I will begin to explain some of these ideas, beginning with
> an overview of what is known and what the goals are, as well as some
> background history.   I will then introduce the so-called p-adically
> completed cohomology spaces attached to reductive groups over number
> fields, which will serve as p-adic analogues of classical spaces of
> modular forms.  If time permits, I will explain the expected relationship
> between these spaces and spaces of Galois representations.
>
>
>

No Langlands seminar on Monday.
Emerton will continue on May 7 (Thursday).

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

Matthew Emerton (Northwestern University)
An introduction to the p-adic Langlands program. III.

No Langlands seminar on Monday.
Emerton will continue on May 14 (Thursday).

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

Matthew Emerton (Northwestern University)
An introduction to the p-adic Langlands program. IV.

1. No Langlands seminar on Monday.
Emerton will continue on May 21 (Thursday).

2. I wonder if there are any volunteers (e.g., among students)
to give a talk in AUTUMN based on the following article:
K.Fujiwara, Independence of $\ell$ for intersection cohomology
(after Gabber), Algebraic geometry 2000, Azumino (Hotaka), 145--151, Adv.
Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002.
(The talk should also include some preliminaries on l-adic cohomology.)

You can download the paper from

      http://www.math.uchicago.edu/~drinfeld/Fujiwara-Gabber.pdf

Here is some information on the article. Please contact me if you need
more informtaion to decide whether you are interested.


                      Subject of the article

Ideally one would like to prove that the categories of $\ell$-adic
sheaves on schemes of finite type over a finite field and the
natural functors between them "don't depend" on $\ell$.
Gabber proved some results in this direction in a simple and beautiful way.

               Pre-requisites for reading the article

To read the article, you have to believe in Lefschetz trace formula and
the existence and properties of the standard functors f_*, f_!, f^*, f^!
in etale cohomology. You also have to know a little bit about perverse
sheaves.

1. Contrary to what I announced before,
there will be NO MORE SEMINAR THIS QUARTER
(in particular, no more talks by Emerton).
I am very sorry for the confusion.

2. I would like to draw your attention to the
Conference and Workshop on Topological Field Theories
      at Northwestern University during the next 2 weeks
(Workshop: May 18-22, 2009; Conference: May 25-29, 2009).

More information about these events can be found at
  http://www.math.northwestern.edu/~pgoerss/tftemphasis/