Beijing Institute of Mathematical Sciences and Applications

Motivated in part by recent work of Hong Liu and Edward Witten etc. on the factorization puzzle and spectral universality in quantum gravity, this topics course will explore the mathematical structures underlying semiclassical analysis and quantum chaos, and the emergence of universal spectral statistics described by random matrix theory. A central theme will be trace formulas, which connect quantum spectra with classical periodic orbits. Periodic-orbit methods also arise more broadly in the study of complex dynamical systems, with connections ranging from analytic number theory to turbulence.

We will develop selected tools from semiclassical and microlocal analysis and use them to study spectral asymptotics, Hamiltonian dynamics, periodic-orbit theory, and trace formulas. We will also explore spectral statistics and random matrix universality, including spectral rigidity and the spectral form factor. Further topics may include the Selberg trace formula and connections between quantum chaos, periodic-orbit theory, and the Riemann zeta function, including selected work of Berry and Keating.

The precise selection of topics will be adjusted according to the pace and interests of the participants. No prior knowledge of semiclassical analysis or quantum chaos will be assumed.

Main references:
M. Zworski, Semiclassical Analysis;
F. Haake, Quantum Signatures of Chaos;
selected papers of M. V. Berry and J. P. Keating.