Nodal sets are zero sets of eigenfunctions of the Laplacian on a Riemannian manifold. Local analysis studies nodal sets in small balls, ignoring the global geometry. Global analysis exploits the dynamics of the geodesic flow to obtain information on nodal sets. First, I will describe the recent proof by Alexander Logunov of Yau's lower bound conjecture for hypersurface volumes of nodal sets. It is a local proof based mainly on the combinatorics of the Donnelly-Fefferman doubling exponent bounds. Second, I will describe recent results on numbers of nodal domains on surfaces of non-positive curvature. These results are based on the ergodicity of the geodesic flow.
We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides.
I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, SL(n,ℤ) acting on 𝕋n, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,ℤ) actions on 𝕋n with standard homotopy data as well as global rigidity of Anosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.
I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions.
The pentagram map and its analogues act on interesting and complicated spaces. The simplest of them is the classical moduli space M0,n of rational curves of genus 0. These moduli spaces have a rich combinatorial structure related to the notion of "Coxeter frieze pattern" and can be understood as a "cluster manifolds". In this talk, I will explain how to describe the action of the pentagram map (and its analogues) in terms of friezes. The main goal is to understand how this action fits with the cluster algebra structure, and in particular, with the canonical (pre)symplectic form.
For a geometrically finite hyperbolic group with small critical exponent, the spectral
method for counting is not available, as there is no point eigenvalue of the Laplace operator on the L2-spectrum. We will explain counting results for orbits of a big class of thin groups acting on a symmetric variety of the real hyperbolic group, which are obtained via ergodic approach.