Normal approximation for recovery of structured unknowns in high dimension: Steining the Steiner formula Larry Goldstein, University of Southern California Abstract Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications. In particular, the discrete probability distribution L(VC) given by the sequence v0,...,vd of conic intrinsic volumes of a closed convex cone C in Rd summarizes key information about the success of convex programs used to solve for sparse vectors, and other structured unknowns such as low rank matrices, in high dimensional regularized inverse problems. The concentration of VC implies the existence of phase transitions for the probability of recovery of the unknown in the number of observations. Additional information about the probability of recovery success is provided by a normal approximation for VC. Such central limit theorems can be shown by first considering the squared length GC of the projection of a Gaussian vector on the cone C. Applying a second order Poincar´e inequality, proved using Stein’s method, then produces a non-asymptotic total variation bound to the normal for L(GC). A conic version of the classical Steiner formula in convex geometry translates finite sample bounds and a normal limit for GC to that for VC. Joint with Ivan Nourdin and Giovanni Peccati. http://arxiv.org/abs/1411.6265

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.