Mathematics Colloquium Spring 2018

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

***Do to Pending Strike: This Colloquium will be added at a later date***

Tuesday, May 8, 2018

Michel Broue, University of Paris 7 & MSRI

Tuesday, May 15, 2018

Sveltana Jitomirskaya, University of California Irvine

Lyapunov exponents, small denominators, arithmetic spectral transitions, and universal hierarchical structure of quasiperiodic eigenfunctions.

A  very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions  has  remained an important open challenge even at the physics level.

I will present a complete solution of this problem in the exponential sense throughout the entire localization regime.  Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature.

These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994.

The talk is based on papers joint with W. Liu.

Tuesday, May 22, 2018

TBA