# 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**

*****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**

**Tuesday, May 29, 2018**

**Chongsheng Cao, Florida International University **

*The global wellposedness problems to some Boussinesq equations*

Oceans and atmosphere are main factors to affect climate dynamics. Boussinesq system is governing equations to the fluid flow of oceans and atmosphere. The system consists of the Navier-Stokes equations and a heat transport equation. It is well known that the global wellposedness of the 3D Navier-Stokes equations is still open. In this talk we will discuss some reduced 3D Boussinesq systems, or simplified models, and also 2D Boussinesq systems, which play important role in understanding our climate. We will present results about the global regularity to these systems.

**Tuesday, June 5, 2018**

**Samit Dasgupta, University of California Santa Cruz**

** Stark's Conjectures and Hilbert's 12th Problem**In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem.