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...
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?
In this talk I will combine...
Birkhoff sections have been invented... by Poincaré in his work
on celestial mechanics. Birkhoff made an extensive use of this
concept in dynamical systems. Sometimes, one can find surfaces
transverse to the trajectories of a vector field in a 3...
We will review some interactions between random matrix theory
and distributions of zeroes of L-functions in families (the
Katz-Sarnak philosophy) before presenting some recent results
(joint with Dorian Goldfeld) in the higher rank setting. We
will...
I discuss a renormalization group method to derive diffusion
from time reversible quantum or classical microscopic dynamics. I
start with the problem of return to equilibrium and derivation of
Brownian motion for a quantum particle interacting with...
The quantum random energy model is a random matrix of
Schroedinger type: a Laplacian on the hypercube plus a random
potential. It features in various contexts from mathematical
biology to quantum information theory as well as an
effective...
Classical matrix perturbation bounds, such as Weyl (for
eigenvalues) and David-Kahan (for eigenvectors) have, for a long
time, been playing an important role in various areas: numerical
analysis, combinatorics, theoretical computer science...
We discuss the classical and non-commutative geometry of wire
systems which are the complement of triply periodic surfaces. We
consider a C∗-geometry that models their electronic properties. In
the presence of an ambient magnetic field, the relevant...
We consider the problem of defining cylindrical contact
homology, in the absence of contractible Reeb orbits, using
"classical" methods. The main technical difficulty is failure of
transversality of multiply covered cylinders. One can fix
this...
I will begin with a brief introduction to the deformation theory
of Galois representations and its role in modularity lifting. This
will motivate the study of local deformation rings and more
specifically flat deformation rings. I will then discuss...
