What is the volume of the set of singular symmetric matrices of norm one? What is the probability that a random plane misses this set? What is the expected "topology" of the intersection of random quadric hypersurfaces?
The study of the Gaussian limit of linear statistics of eigenvalues of random matrices and related processes, like determinantal processes, has been an important theme in random matrix theory. I will review some results starting with the strong Szegö limit theorem, and also discuss the possibility of non-Gaussian limits.
I will discuss recent joint work with Vinicius Gripp and Michael Hutchings relating the volume of any contact three-manifold to the length of certain finite sets of Reeb orbits. I will also explain why this result implies that any closed contact three-manifold has at least two embedded Reeb orbits.
This workshop is part of the topical program "Non-equilibrium Dynamics and Random Matrices" which will take place during the 2013-2014 academic year at the Institute for Advanced Study.
The workshop aims to provide a broad survey of the current status of Non-equilibrium Dynamics and Random Matrices. The main topics addressed in this workshop are the mathematical foundations of random matrix theory and the random Schrodinger equation, and the applications of random matrix theory to physics, combinatorics, and engineering.