Jens Marklof
University of Bristol
January 22, 2025
Quantum chaos, moduli spaces and random matrix theory.: One of the long-standing conjectures in quantum chaos is that the spectral statistics of quantum systems with chaotic classical limit are governed by random matrix theory. Despite convincing heuristics, there is currently not a single example where this phenomenon can be established rigorously. Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this colloquium we will review Rudnick's approach and extend it to explain the emergence of the Gaussian Unitary Ensemble for twisted Laplacians (which break time-reversal symmetry) and to the Gaussian Symplectic Ensemble for Dirac operators. This lecture is aimed at a broad mathematical audience and based on joint work with Laura Monk (Bristol).
University of Bristol
January 22, 2025
Quantum chaos, moduli spaces and random matrix theory.: One of the long-standing conjectures in quantum chaos is that the spectral statistics of quantum systems with chaotic classical limit are governed by random matrix theory. Despite convincing heuristics, there is currently not a single example where this phenomenon can be established rigorously. Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this colloquium we will review Rudnick's approach and extend it to explain the emergence of the Gaussian Unitary Ensemble for twisted Laplacians (which break time-reversal symmetry) and to the Gaussian Symplectic Ensemble for Dirac operators. This lecture is aimed at a broad mathematical audience and based on joint work with Laura Monk (Bristol).