# Mathematics Colloquium Spring 2018

For further information please contact Professor Torsten Ehrhardt or call the Mathematics Department at 459-2969

**Tuesday, April 3, 2018**

**Julia Pevtsova, University of Washington & MSRI**

**Supports and tensor ideals in modular representation theory**

Modular representation theory studies representations of a finite group and other algebraic structures such as Lie algebras over a field of positive characteristic. Classifying modular representation up to direct sums – as the classical theory does for complex representations – is often a hopeless task even for such a tiny group as Z/3 x Z/3. I’ll discuss a geometric approach to understanding this wild territory starting with Quillen’s classical work on group cohomology and leading to the applications of the ideas in tensor triangular geometry of P. Balmer in the context of modular representation theory.

**Tuesday, April 10, 2018**

**Mimi Dai, University of Illinois at Chicago**

**Regularity problem for the Navier-Stokes equation and beyond**

The Navier-Stokes equation (NSE) is a fundamental model that describes the motion of turbulent flows. Leray’s conjecture in 1934 regarding the appearance of singularities of solutions to the 3-dimensional NSE remains open. I will start from here and continue to introduce some recent progress in this area by using paradifferential calculus and harmonic analysis techniques. Of particular interest are some new findings on conditional regularity and determining modes for the NSE, based on the recently developed wavenumber splitting framework.

**Tuesday, April 17, 2018**

**Jacques Thevenaz, Ecole Polytechnique federal de Laussanne & MSRI**

**Linear representations of finite sets**

(Joint work with Serge Bouc.) A linear representation of a finite set X is a vector space V together with linear operators from V to V, one for each relation on X, subject to elementary multiplicative conditions. Recall that a relation on X is a subset of the direct product X x X (also called a Boolean matrix).

Among all such linear representations, there is the question of finding the dimensions of the simple ones. This is not easy and remained an open problem for a long time. We solve this problem by embedding this representation theory into a larger one, namely the linear representations of all finite sets. By this we mean a vector space V(X) for each finite set X together with linear maps from V(X) to V(Y), one for each correspondence between X and Y. Recall that a correspondence between X and Y is a subset of X x Y.

We will give an introduction to this larger representation theory, which is very rich. In particular, we will explain how to find the dimensions of the simple representations.

** NEW Speaker...Tuesday, April 24, 2018**

**Burkhard Kuehlshammer, University of Jena, visiting UCSC**

**Around Donovan's Conjecture in Representation Theory**

For a fixed prime number p, the irreducible representations of a finite group G can be partitioned into so-called "p-blocks". It was observed by R. Brauer in the sixties that a considerable part of such a p-block B is already determined by a relatively small subgroup of G, the "defect group" D of B. This somewhat vague observation was given a rigorous mathematical setting by P. Donovan in the seventies. The corresponding Donovan Conjecture is, however, still open. It is one of the main open problems in modular representation theory of finitegroups. In this colloquium talk, I will report on some of the developments related to this conjecture and on some recent progress.

**Tuesday, May 1, 2018**

**TBA**

**Tuesday, May 8, 2018**

**Michel Broue, University of Paris 7 & MSRI**

**Tuesday, May 15, 2018**

**Sveltana Jitomirskaya, University of California Irvine**

**Tuesday, May 22, 2018**

**TBA**

**Tuesday, May 29, 2018**

**TBA**

**Tuesday, June 5, 2018**

**TBA**