Geometric Langlands Seminar

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



  • Date: Fri, 26 Sep 2008 13:38:00

  • 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.
    
    
    
    
    
    
    
    


  • Date: Wed, 1 Oct 2008 09:28:32

  • 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).
    
    
    
    
    
    
    


  • Date: Wed, 1 Oct 2008 11:27:41

  • 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).
    >
    >
    >
    >
    >
    >
    
    
    
    


  • Date: Fri, 3 Oct 2008 09:32:38

  • 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).
    
    
    
    
    
    
    
    
    
    
    


  • Date: Wed, 8 Oct 2008 10:01:33

  • 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



  • Date: Fri, 10 Oct 2008 12:58:41

  • 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).