In this talk, I will be discussing various funding and other opportunities for mathematical and statistical sciences at NSF. New programs in computational and data-enabled sciences will be discussed. Also,this is an opportunity to hear feedback from the community regarding future needs.

Discrete quasiperiodic Schrodinger operators have been researched extensively over the past thirty years to produce a rather complete spectral analysis when the potential is defined by analytic functions. However, the nature of the spectral measures for less than $C^\infty$ regularity of the potential is largely unknown. We demonstrate that, with only minimal assumptions on the regularity of the potential, in the regime of positive Lyapunov exponents, the spectral measures are always of
Hausdorff dimension zero.

We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint work with Amir Dembo, Tomoyuki Ichiba, Soumik Pal and Ofer Zeitouni