I will report on recent progress towards fundamental conjectures in quantum chaos, which assert that the statistics of quantum energy levels should be governed by a Poisson point process if the system is completely integrable (Berry & Tabor, 1977) or by random matrix theory if the underlying classical dynamics is “chaotic” (Bohigas, Giannoni & Schmit, 1984). In the case of the Laplacian on flat tori, the question reduces to a natural problem on the distribution of values at integers of a quadratic form, which in turn is intimately related to the classical Oppenheim conjecture. We will also discuss the Laplacian on a compact hyperbolic surface and review some recent evidence for the emergence of random matrix statistics in the limit of high genus. This talk will straddle aspects of analysis, geometry, number theory as well as mathematical physics, and should be accessible to a broad audience.
Zoom link: https://icts-res-in.zoom.us/j/91031372314?pwd=YcCNCSguq7Ta4aa5Mi2w2Cpd0Zcavz.1
Meeting ID: 910 3137 2314
Passcode: 202040