In the 1980's (at UWO) I gave a local construction of the Deligne-Langlands epsilon factors attached to representations of Galois groups of local field extensions. The method was to resolve an arbitrary representation by monomial representations, for which the construction was straightforward. At the time my idea was to attack the Langlands programme by making a similar resolution of an arbitrary admissible representation of $GL_{n}K$ where $K$ is a local field.

Returning to this with a bit more knowledge, I now more or less have the correct definition and the outline of the construction to the extent that I can handle $GL_{2}K$!

The entire Langlands programme has many features which were suggested by properties of representations of finite groups such as $GL_{n}{\mathbb F}_{q}$.

So I shall spend a lot of the time illustrating the constructions and properties in the case of finite groups - looking at some or all of: (i) Weil representations, (ii) cuspidality and monomial resolutions, (iii) local L-functions and (iv) Shintani descent.

Several models for $(\infty, 1)$-categories have been defined and shown to be equivalent, and they are all being used in different areas of algebra and topology. More recently, there has been interest in more general $(\infty, n)$-categories, especially with Lurie's recent work on the Cobordism Hypothesis. Comparison of different definitions is still work in progress by several authors. In this talk, we will go over some of the models for $(\infty, 1)$-categories and discuss some of the methods for inductively generalizing them to models for $(\infty, n)$-categories.

In this talk, we will construct a kernel that will give a new characterization of doubly stochastic operators. This is needed in the proof of the Horn theorem.

I will introduce the Heisenberg group, and will elaborate on the Stone-von Neumann Theorem which characterizes the unitary dual of this group. I will then prove the Plancherel Theorem for the Heisenberg group.

The theory of group-valued moment maps provides a natural framework for moduli spaces of flat G-bundles over surfaces. In this talk, I will describe a quantization procedure for such moment maps. An application to the moduli space example gives the Verlinde numbers.

For any given field $F$ there is a well known parametrizing space for elementary p-abelian Galois extensions of $F$; for example, if $K$ contains a primitive pth root of unity, Kummer theory provides this parametrizing space for us. By putting additional structure on these parametrizing spaces, we are able to give a parametrizing space for solutions to any given embedding problem where the quotient is a cyclic $p$-group and the kernel is an elementary $p$-abelian group. This allows us to give an explicit count to the number of such solutions, and in particular we can make certain universal statements about the number of solutions to such embedding problems. For instance, we use our results to show that $p$-groups have unbounded realization multiplicity.