Let $\g$ be a reductive Lie algebra. For any two crystals $A$ and $B$ of $\g$ representations, $A \otimes B$ and $B \otimes A$ are isomorphic, but not by the obvious map $a \otimes b \rightarrow b \otimes a$. In a previous work, Henriques and Kamnitzer construct a natural isomorphism $\sigma_{A, B} : A \otimes B \rightarrow B \otimes A$, called the commutor, which has many nice properties. Their definition uses Schutzenberger involution, and is particular is only defined for $\g$ of finite type. In this work we give a new definition of the commutor. This version relies on Kashiwara involution, and is defined for any symmetrizable Kac-Moody algebra. This will be a short talk, which will consist mostly of pictures.

Suppose that $G$ is a finite group of size $n$, and $V$ is an irreducible representation of dimension $d$.  Certainly, $d^2 \leq n$, but how close can $d^2$ be to $n$?  In particular, if $n = d(d+e)$ and we fix $e$, what can be said about $G$ and $V$?  I'll classify all examples with $e=1$ and show that there are only finitely many examples with any other fixed $e$.

I plan to give a series of talks on the circle of ideas relating the representation theory of semi-simple Lie algebras, the geometry of the flag variety, and Kazhdan-Lusztig polynomials.  I'll work on as basic a level as possible, pitching toward those who've finished 256B and 261B.  Hopefully this series can serve well as a complement to the
seminar on geometric Langlands.

In the first talk, I'll recall the basics of category O, what it is, how it is structured and, in particular, its block decomposition.  As time permits, I'll also discuss D-modules on the flag variety, and the localization theorem of Beilinson and Bernstein relating them to representations of a semi-simple Lie algebra.

In this lecture, we will finally get to the geometry part. I'll cover the localization theorem of Beilinson and Bernstein, and the Riemann-Hilbert correspondence, hopefully making it through to the definition of perverse sheaves.

I'll go back over the localization theorem of Beilinson and Bernstein, which I covered with undue celerity in the last lecture, which shows that a block of category O is equivalent to the category of B-equivariant D-modules on G/B. Then we'll progress to using this theorem show how coefficients of character formulae in category O can be calculated by counting points, via the Lefschetz fixed point formula.

In the last talk of this series, I will discuss how the Weil conjectures and the theory of weights can be applied to solve all our problems. Assuming the only problems we have are relating character formulae in category O to the structure of the Hecke algebra.

There's some emerging folk wisdom which says that there's a connection between the "n" in n-category and the "n" in n-dimensional. In this talk, we'll explore one of the simplest examples of this phenomenon, showing how one can recover the axioms of a (finite) quantum group from path integral rules of a "categorically enriched" version of the topological QFT known as Chern-Simons theory. The finiteness restriction is necessary in this case to make the path integral into a well-defined mathematical object.

Notes from this talk are available at http://math.berkeley.edu/~ajt/qgpi.pdf

Last week, we saw that a QFT can be described as a monoidal functor from a monoidal category of d-dimensional bordisms to the category of vector spaces. We also introduced Dan Freed's improved definition, which describes a quantum field theory as a monoidal functor from a category of bordisms carrying fields to the category of one-dimensional Hilbert spaces. This week, we will extend this definition in categorical fashion, by looking at functors from the 2-category of bordisms and "bordisms between bordisms" to a 2-category of categorified Hilbert spaces. Then we'll work through a single example, recovering the quantum double of the functions on a finite group G from Freed's description of the QFT of maps from a 3-manifold to BG.

In this talk, we will review some interesting applications of modular forms in topology. The modular invariance techniques lead to some cancelation formulas for characteristic forms and consequently imply divisibility results of some characteristic number, e.g. the signature of spin manifolds. We also discuss the odd dimensional analogues, which heuristically study the Chern-Simons transgression on loop space.

I'll introduce the 6-vertex model and show how R matrices for evaluation representation of $U_q(\widehat{sl_2})$ can be used to express the one point functions for this model. I'll explain how the Yang-Baxter equation can be used to show the commutativity of column transfer matrices and how Baxter's corner transfer matrix method can be used to calculate the one point functions for this model.

We will define and give examples of chiral algebras and factorization algebras, pointing their connection to the representation theory of Lie algebras. If time permits, we will say a few words as to why these objects are useful and/or interesting (eg. Geometric Langlands).

The symmetric group S_n acts on the polynomial ring Q[x_1,...,x_n] by permuting the variables. The decomposition of this representation into irreducibles is well-known, but it is not well understood how this decomposition is compatible with multiplication in the polynomial ring. Garsia and Procesi give a partial answer that involves the q-Kostka polynomials, and there seems to be a lot more nice combinatorics hiding in this problem. I'll discuss a work in progress that attempts to solve this problem by generalizing it to the Hecke algebra of S_n and looking at canonical bases there. In particular, I'll describe Kazhdan-Lusztig cells of a Hecke algebra representation that at q=1 is the tensor algebra of the defining representation of S_n.

The goal of this talk is to enable representation theory students to start attending Subfactor seminar without being confused for the first month or two. I'll take a very algebraic approach, and will avoid analytic concepts like ``weak operator closures" and ``$L^2$." I'll talk a little bit about what a Subfactor is, go over the basic construction, talk about planar algebras, and say a little bit about where examples come from.