Georgia Institute of Technology College of Sciences

Much effort in the past several decades has gone into lifting various algebraic structures into a topological context. I will describe one such lifting: that of the arithmetic theory of elliptic curves. The result is a rich and highly structured family of cohomology theories collectively known as elliptic cohomology. By forming "global sections" one is led to a topological enrichment of the ring of modular forms. Geometric interpretations of these theories are enticing but still conjectural at best.

Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed:the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss a result showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.

In this presentation I will show some of the surprising depth and complexity of elementary- and middle-grades mathematics, much of which has been revealed by detailed studies into how students think about mathematical ideas. In turn, research into students' thinking has led to the development of teaching-learning paths at the elementary grades, which are reflected in the Common Core State Standards for Mathematics. These teaching-learning paths are widely used in mathematically high-performing countries but are not well understood in this country. At the middle grades, ideas surrounding ratio and proportional relationships are critical and central to all STEM disciplines, but research is needed into how students and teachers can reason about these ideas. Although research in mathematics education is necessary, it is not sufficient for solving our educational problems. For the mathematics teaching profession to be strong, we need a system in which all of us who teach mathematics, at any level, take collective ownership of and responsibility for mathematics teaching.

Random matrix theory (RMT) is a very active area of research and a greatsource of exciting and challenging problems for specialists in manybranches of analysis, spectral theory, probability and mathematicalphysics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to the concepts of determinantal point processes and to their particular case, biorthogonal ensembles, when the main object to study, the correlation kernel, can be written explicitly in terms of two sequences of mutually orthogonal functions.Another source of determinantal point processes is a class of stochasticmodels of particles following non-intersecting paths. In fact, theconnection of these models with the RMT is very tight: the eigenvalues of the so-called Gaussian Unitary Ensemble (GUE) and the distribution ofrandom particles performing a Brownian motion, departing and ending at the origin under condition that their paths never collide are, roughlyspeaking, statistically identical.A great challenge is the description of the detailed asymptotics of these processes when the size of the matrices (or the number of particles) grows infinitely large. This is needed, for instance, for verification of different forms of "universality" in the behavior of these models. One of the rapidly developing tools, based on the matrix Riemann-Hilbert characterization of the correlation kernel, is the associated non-commutative steepest descent analysis of Deift and Zhou.Without going into technical details, some ideas behind this technique will be illustrated in the case of a model of squared Bessel nonintersectingpaths.