We consider Lie groups, Lie algebras, and their categories of representations. These structures can be deformed, leading to the quantum groups $U_q(g)$. Representations of $U_q(g)$ form a braided monoidal category, where the braiding is constructed using a universal $R$-matrix. We will discuss what this means, why it is useful, and perhaps even how one constructs the $R$-matrix. This is our first talk of the semester, and should be widely accessible.

The main goal of this talk is to motivate the deformation we saw last week by relating it to actual physics. A Quantization of the algebra of functions on a Lie group can be used to define a Poisson-Lie group structure. In fact, this argument should be turned around: one should start with a Poisson Lie group, then try to quantize. In this form, our construction is analogous to the ``deformation quantization" formulation of quantum mechanics. Despite the introduction of physics, I hope to keep this accessible.

The aim of this talk is to construct a (projective) quantum field theory using geometric quantization. The main ingredient is the construction of a Hilbert space associated to a surface. At first this appears to depend on a choice of complex structure, but we will see that, at least projectively, it does not. This will be done without any discussion (on my part, at least) of the physical aspects or use of local coordinates.

In this talk I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by "fake quantum dimensions." In the case of simple Lie algebras these "fake quantum dimensions" are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of links. This is joint work with Bertrand Patureau-Mirand and Vladimir Turaev.

One of the key concerns of mathematics is classification theorems. I'll tell you how to classify abelian categories (with a few niceness conditions), and (as time and audience permits) say a little bit about why this approach doesn't work so well (with fascinating results) in the case of triangulated categories (i.e. derived categories). Algebraic geometry, quivers, and the Freyd embedding theorem may make special appearances as necessary.

In this talk, I'll use some heavy machinery to compute the characters of the irreps of $SU(2)$, in a manner which will hopefully explain the ideas behind the heavy machinery. More precisely: We will set up the Borel-Weil correspondence between representations of $SU(2)$ and line bundles on the flag variety $X_{SU(2)} \simeq \mathbb{P}^1$. Then we will think of such line bundles as living in the equivariant K-theory ring $K_{U(1)}(\mathbb{P}^1)$, and we will study the "integration" map $K_{U(1)}(\mathbb{P}^1) \to K_{U(1)}(\operatorname{pt})$. The latter map will turn out to send line bundles to their characters, and we will compute these characters by localizing to the fixed points of $U(1)$ on $\mathbb{P}^1$.

The colored HOMFLY polynomial is a quantum invariant of oriented links in $S^{3}$ associated with a collection of irreducible representations of each quantum group $U_{q}(sl_{N})$ for each component of the link. We will discuss in detail how to construct these polynomials and their general structure, which is the part of Labastida-Marino-Ooguri-Vafa conjecture. The new integer invariants are also predicted by the LMOV conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory.

I plan to explain relationship between two approaches to representation theory of gl(m|n), geometric approach involving Borel-Weil-Bott theorem on supervarieties, developed by I. Penkov and myself, and algebraic approach involving canonical bases for gl(\infty) developed by J. Brundan. These methods produce different combinatorial algorithms for calculating Kazhdan-Lusztig polynomials. For some time equivalence of these algorithms was not known. Recently I. Musson and I proved it using a beautiful notion of weight diagrams invented by J. Brundan and C. Stroppel. The main part of the talk will concern these diagrams, it will be purely combinatorial.

Affine Lie algebras and quantum groups are two different generalizations of simple Lie algebras. The study of solutions to certain differential equations called Knizhnik-Zamolpdchikov equations unites the representation theory of affine Lie algebras and quantum groups. In this talk I will explain the KZ equations, focusing mainly on the first step, which is the construction of the operator KZ equations.

Given a Young diagram, there is a simple formula, called the hook formula, giving the number of possible standard tableaux on it. This formula can be proved by a simple probabilistic argument which I will present. The basic idea is the following: To prove that a certain sum is equal to 1, we construct a random process (called the hook walk) with probabilities equal to the terms in the sum. This idea can be generalized to a q-version which I will discuss in detail, and the machinery also provides a random process for constructing Young tableaux (of a given shape or size) with certain prescribed probabilities.

We introduce a measure on strict plane partitions that is an analog of the uniform measure on plane partitions. We describe this measure in terms of a Pfaffian point process and compute its bulk limit when partitions become large. The above measure is a special case of the shifted Schur process, which generalizes the shifted Schur measure introduced by Tracy and Widom and is an analog of the Schur process introduced by Okounkov and Reshetikhin. We prove that the shifted Schur process is a Pfaffian point process and calculate its correlation kernel. Along the way we also obtain a one parameter generalization of Macmahon's formula for the generating function of plane partitions.