Graduate students may wonder how the muscle power of mathematics can be used to solve, or at least shed light, on serious concerns from other disciplines.  In this expository presentation, I offer some examples.  The first is how orbits of symmetry groups can resolve a range of long-standing puzzles coming from voting to statistics to …  A second is how related ideas introduce new insights about the compelling “dark matter” mystery from astronomy.

In 1932 von Neumann proposed classifying the statistical behavior of measure preserving diffeomorphisms of the torus, In joint work with B. Weiss I prove this is impossible (in a convincing and rigorous sense) even in the simplest and most concrete case: the 2-torus. 

By luck our work has accidental, but far-reaching consequences inside ergodic theory.  It establishes a “global structure theorem” for ergodic measure preserving transformations that gives heretofore unknown and surprising examples of diffeomorphisms of the torus.

I will discuss three different contexts in which commutative
rings of functions and modules over them are replaced by their
non-commutative versions. One is the ring of differential operators, where
the modules correspond to systems of differential equations. The second
setting is geometric quantization which provides a baby version of the
Hilbert space in quantum mechanics. The third setting is deformation
quantization in symplectic geometry. I will explain a relation between these
three versions, although the reasons behind the relations are not quite well
understood.

The origins of Carleman estimates lie with the pioneering 1939 work by the Swedish mathematician T. Carleman, concerned with the unique continuation property for solutions for linear elliptic partial differential equations with smooth coefficients in dimension two. The fundamental new idea introduced by Carleman, which consists of establishing a priori energy estimates involving an exponential weight, has permeated essentially all the subsequent work in the subject. More recently, Carleman estimates have found numerous striking applications beyond the original domain of unique continuation, from control theory to spectral theory to the analysis of inverse problems. The purpose of this talk is to provide a broad introduction to the subject and to attempt to illustrate some of its inner workings.

This will be a general lecture introducing the spacetime of relativity.
Most discussions will concern the Minkowski spacetime (flat space) and the
Schwarzschild spacetime, but we will try to hint at the nature of the Einstein
equations and how they determine spacetime from initial data.