We will discuss the exponential map (from a Lie algebra to the corresponding Lie group) in the case of positive characteristic p, and its relation to the Campbell-Baker-Hausdorf formula which expresses the group product via the Lie brackets. If time permits, we will also talk about loops (algebraic structures similar to groups where only a weaker form of associativity holds).

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.

The period and index of a curve C are two quantities which describe the failure of C to have rational points. The mismatch between the two is of interest for its impact on the Shafarevich-Tate group of the Jacobian of C. The period-index problem is to determine what values of period and index are possible for a given genus g. We will give a complete answer when g=1, and an almost complete answer when g ≥ 2.