In this talk we relate concentration of Laplace eigenfunctions in position and momentum to sup-norms and submanifold averages. In particular, we present a unified picture for sup-norms and submanifold averages which characterizes the concentration of those eigenfunctions with maximal growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. 

The most fundamental kind of functions studied in computer science are Boolean functions. They take n bits as an input and return one bit as an output. Most Boolean functions oscillate a lot, which is analogous to the fact that "most" continuous functions on R are nowhere differentiable. If we want to generate a "smooth" Boolean function, we can take the sign of some polynomial of low degree in n variables. Such functions are called polynomial threshold functions, and they are widely used in machine learning as classification devices. Surprisingly, we do not know how many polynomial threshold functions there are with a given degree! Even an approximate answer to this question has been known only for polynomials of degree 1, i.e. for linear functions. In a very recent joint work with Pierre Baldi, we found a way to approximately count polynomial threshold functions of any fixed degree. This solves a problem of M. Saks that goes back to 1993 and earlier. Our argument draws ideas from analytical number theory, additive combinatorics, enumerative combinatorics, probability and discrete geometry. I will describe some of these connections, focusing particularly on a beautiful interplay of zeta and Mobius funcitons in number theory, hyperplane arrangements in enumerative combinatorics and random tensors in probability theory.

This is the first of two talks about perfectoid algebras, a concept which was developed in the last 10 years by Peter Scholze.  The talks are intended to be introductory.  Some familiarity with the p-adic numbers would be helpful but shouldn't be necessary.