Activities This Week

In a seminal series of works, culminating in the monumental “What Determines an Algebraic Variety?” János Kollár, Max Lieblich, Martin Olsson, and Will Sawin prove that a normal projective algebraic variety of dimension at least 2 over an uncountable field of characteristic 0 can be reconstructed, in a precise sense, solely from its underlying topological space. The results of KLOS are specific to char. 0 and to normal varieties. Castle and O’Gorman, using the model theoretic machinery of Zilber’s Restricted Trichotomy, extend these results to all quasi projective varieties (of dimension at least 2) in all characteristics, in the case where the underlying field is algebraically closed and uncountable. In the talk, I will present the results and try to sketch the strategy of proof of the new result.

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak says that the L^2 mass of eigenfunctions of the Laplacian in hyperbolic manifolds equidistributes, as the eigenvalues tend to infinity. We consider a special class of such functions, Hecke—Maass forms, that are central in number theory. The conjecture has been established for these functions in dimension 2 and 3, but in dimension 4 there is a new challenge: one needs to rule out concentration of measure along certain large totally geodesic submanifolds. We will discuss our recent result in which we overcome this difficulty for a particular sequence of eigenfunctions known in number theory as lifts. This is a joint work with Alexandre de Faveri.

We survey progress from the past five years on the distribution of mass in high-dimensional convex bodies and in probability distributions with convexity properties. The concentration of measure phenomenon has traditionally been studied in highly regular or structured settings, such as spheres, Hamming cubes, Gaussian measures, Markov chains, and martingales. It turns out that convexity assumptions provide an alternative source of regularity in high dimensions with remarkably similar features: Lipschitz functions are highly concentrated, the isoperimetric problem is nearly saturated by half-spaces (up to logarithmic factors), and the central limit theorem is nearly as strong as in the setting of independent random variables. The main developments discussed include the resolution of Bourgain’s slicing problem and the Variance Conjecture, as well as recent progress on the isoperimetric problem for high-dimensional convex bodies. Based on joint work with P. Bizeul and J. Lehec.