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).