# Antti Knowles receives a ERC Starting grant

Assistant professor Antti Knowles received a ERC Starting Grant from the European Research Council (ERC). This new grant RandMat adds to the four ERC Advanced grants currently held at the Section of Mathematics by Professors Anton Alekseev (MODFLAT) Stanislav Smirnov (COMPASP), Grigory Mikhalkin (TROPGEO) et Jean-Pierre Eckmann (BRIDGES).

## RandMat - Spectral statistics of structured random matrices

The purpose of this project is to answer fundamental questions about several important classes of random matrices, using techniques from analysis, probability, combinatorics, and theoretical physics. The past decade has seen several breakthroughs in our understanding of the distribution of eigenvalues and eigenvectors of large random matrices. Much of this effort has focused on mean-field random matrices with independent entries. Instead, this project proposes to investigate random matrices with some nontrivial structure that arises naturally in important applications and leads to a rich mathematical behaviour. The two main classes are:

- random graphs with fixed degrees, such as random regular graphs,
- random band matrices, which constitute a good model of disordered quantum Hamiltonians.

The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). These results will address several of the salient questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.