L6

Dehn surgery is a method of building three-dimensional manifolds that is ubiquitous throughout low-dimensional topology. I will give an introduction to Dehn surgery and discuss recent work with M. Kegel on the uniqueness of Dehn surgery descriptions of 3-manifolds. To do this, I will discuss the reason that Dehn surgery is so prominent - namely that it interacts very well with many structures, such as the geometry and gauge theory of 3-manifolds. (I will do my very best to assume very little background knowledge.)

A fundamental phenomenon in the representation theory of finite and compact groups is that irreducible characters tend to take smaller values on elements that are far from central. Character estimates of exponential type (that is, bounds of the form |chi(g)|<chi(1)^(1-epsilon)) are particularly useful for probabilistic applications, such as bounding the mixing time of random walks supported on conjugacy classes.

In 1981, Diaconis and Shahshahani established sharp estimates for irreducible characters of the symmetric group S_n, evaluated at a transposition t = (i j). As an application, they proved that roughly n*log(n) random transpositions are required to mix a deck of n playing cards. This was extended in 2007 by Muller--Schlage-Puchta to to arbitrary permutations in S_n. Exponential character bounds for finite simple groups were subsequently developed through a series of works by Bezrukavnikov, Liebeck, Shalev, Larsen, Guralnick, Tiep, and others. 

In this talk, Itay Glazer (Technion) will present recent progress on exponential character estimates for compact Lie groups.

This is based on joint work in progress with Nir Avni, Peter Keevash, and Noam Lifshitz.

Yotam Hendel will discuss how polynomial maps can be studied by examining the analytic behavior of pushforwards of regular measures under them over finite and local fields. 

The guiding principle is that bad singularities of a map are reflected in poor analytic behavior of its pushforward measures. Yotam will present several results in this direction, as well as applications to areas such as counting points over finite rings and representation growth. 

Based on joint work with I. Glazer, R. Cluckers, J. Gordon, and S. Sodin.

An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially discrete series representations of a connected reductive p-adic group is stable under twist by automorphisms of the complex numbers, and hence this class is defined over $\bar{\mathbb{Q}_\ell}$. Recent work of Solleveld, building on work of Kazhdan-Varshavsky-Solleveld, says that the same is true of the class of standard representations. Stefan Dawydiak will give a geometric proof of this result for the principal block, and use this to deduce a local proof of Clozel's theorem for the general linear group. Time permitting, Stefan will also give geometric formulas for certain Harish-Chandra Schwartz functions that help illustrate these results.

A d-dimensional TQFT is a topological invariant which assigns (d-1)-dimensional manifolds to vector spaces and d-dimensional cobordisms to linear maps. In the early 90s, Reshetikhin and Turaev constructed examples of these in the case d=3, using the data of certain types of linear categories. In this talk, I will provide an overview of this construction, and then explore how this might be meaningfully extended downwards to assign 1-manifolds to "2-vector spaces". Minimal knowledge of category theory assumed!

An important invariant in the complex representation theory of reductive p-adic groups is the wavefront set, because it contains information about the character of such a representation. In this talk, Mick Gielen will introduce a new invariant called the canonical dimension, which can be said to measure the size of a representation and which has a close relation to the wavefront set.  He will then state some results he has obtained about the canonical dimensions of compactly induced representations and show how they teach us something new about the wavefront set. This illustrates a completely new approach to studying the wavefront set, because the methods used to obtain these results are very different from the ones usually used.