In recent years, there have been major breakthroughs on the topic of super-approximation for algebraic groups, which is a qualitative version of strong approximation. Super-approximation has proven to be incredibly useful in many areas of both pure and applied Mathematics. We discuss the difficulties of super-approximation in positive characteristic, as well as recent new results for absolutely almost simple groups over $k(t)$, where $k$ is a finite field.

In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogeneous Boitzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

We study the spectrum of complete non-compact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. As applications, we construct metrics with an arbitrarily large finite number of gaps in its essential spectrum on non-compact covering of a compact manifold and complete non-compact manifold with bounded curvature and positive injectivity radius.This is a joint work with Richard Schoen.

Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by uncertainty in their parameters. The polynomial chaos method is a computational approach to solve stochastic partial differential equations (SPDE) by projecting the solution onto a space of orthogonal polynomials of the stochastic variables and solving for the deterministic coefficients. Polynomial chaos can be more efficient than Monte Carlo methods when the number of stochastic variables is low, and the integration time is not too large. When performing long-term integration, however, achieving accurate solutions often requires the space of polynomial functions to become unacceptably large. This talk will introduce alternative approach, where sets of empirical basis functions are constructing by examining the behavior of the solution for fixed values of the random variables. The empirical basis functions are evolved over time, which means that the total number of basis functions can be kept small, even when performing long-term integration.

We will prove the Cayley-Hamilton theorem for commutative rings in an elementary/remarkable way. This is an exposition of a paper by Howard Straubing.

I will talk about the conjecture of Hausel, Letellier and Villegas, which gives precise predictions for mixed Hodge polynomials of character varieties. In certain specializations this conjecture also computes Hurwitz numbers, Kac's polynomials of quiver varieties, and zeta functions of moduli spaces of Higgs bundles. The subject is at an exciting intersection of number theory, algebraic geometry, combinatorics and mathematical physics, and is an area of active research. The talk is deemed as an introduction for a general audience. If time permits, I will explain my recent results in this area.

K3 surfaces are two dimensional Calabi-Yau manifolds. Their moduli space is
of interest in algebraic geometry, but also has connections with number
theory and string theory. I will discuss ongoing joint work with Alina
Marian and Rahul Pandharipande aimed at studying the tautological ring of
the moduli space of K3 surfaces. In particular, I will discuss different
notions of tautological classes. Next, I will explain a method of deriving
relations between tautological classes via the geometry of the relative
Quot scheme.

The proof of Fermat's Last Theorem established a deep relation
between elliptic curves and modular forms, mediated by an equality of
L-functions (which are analogous to the Riemann zeta function). The
common ground is Galois representations, and Wiles' overall strategy
was to parametrize their deformations via algebras of Hecke
operators. In higher rank the global Langlands conjecture posits a
correspondence between n-dimensional Galois representations arising
from the cohomology of algebraic varieties and certain so-called
automorphic representations of $GL(n)$, which belong in the realm of
harmonic analysis. There is a known analogue over local fields (such
as the p-adic numbers $Q_p$) and one of the key desiderata is
local-global compatibility. This naturally leads one to speculate
about the existence of a finer "p-adic" version of the local Langlands
correspondence which should somehow be built from a "mod p" version
through deformation theory. Over the last decade this picture has been
completed for $GL(2)$ over $Q_p$, and extending it to other groups is a
very active research area. In my talk I will try to motivate these
ideas, and eventually focus on deformations of smooth representations
of $GL(n)$ over $Q_p$ (or any p-adic reductive group). It seems to be an
open problem whether universal deformation rings are Noetherian in
this context. At the end we report on progress in this direction
(joint with Julien Hauseux and Tobias Schmidt). The talk only assumes
familiarity with basic notions in algebraic number theory.

This presentation will focus on (1) teaching positions in community colleges and (2) tenure-track faculty positions in universities. The tenure system (as practiced at UCSD) will also be explained.

BONUS: A guest has been invited to talk firsthand about the experience of recently becoming a tenure-track faculty member at a research university.