Title: Arithmetic Quantum Chaos
Abstract: Understanding the semiclassical limit of the quantization of even the simplest classically chaotic Hamiltonian, proved to be problematic from the inception of quantum mechanics 100 years ago. Arithmetically defined such systems provide instances for which this limit can be analyzed mathematically using central tools from number theory and homogeneous dynamics.As such these can be thought of as "solvable models ". We review some of the many developments for these quantizations.
About the Speaker: Professor Peter Sarnak is a South African/American mathematician who has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study since 2007. He is also Eugene Higgins Professor of Mathematics at Princeton University since 2002. Professor Sarnak has made important contributions to analysis and number theory. Sarnak’s work on the existence of cusp forms led to the disproof of a conjecture of Selberg. He has obtained the strongest known bounds towards the Ramanujan–Petersson conjectures for sparse graphs, and he was one of the first to exploit connections between certain questions of theoretical physics and analytic number theory. Professor Sarnak has also made fundamental contributions to arithmetical quantum chaos, a term which he introduced, and to the relationship between random matrix theory and the zeros of L-functions. His work on sub-convexity for Rankin–Selberg L-functions led to the resolution of Hilbert's eleventh problem. Professor Sarnak has received many international prizes in recognition of his broad and deep contributions to mathematics. He was awarded the 2024 Shaw Prize for "his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, analysis, combinatorics, dynamics, geometry and spectral theory”. Professor Sarnak has mentored over 57 PhD students during his career, many of whom have gone to have spectacular careers in mathematics.
This lecture is part of the discussion meeting "A Hundred Years of Quantum Mechanics"