Abstracts SMS 2023 2

We will briefly review the main concepts and fascinating physics background of one-dimensional analytic quasiperiodic operators and related cocycles. We then present the highlights of our joint work with Lingrui Ge, Jiangong You, and Qi Zhou on a non-commutative generalization of the classical Jensen’s formula, and explain how it helps us to uncover the mystery of Avila’s global theory and prove a number of global spectral corollaries.

The hot spots conjecture of J. Rauch states that, given a generic heat distribution on some insulated domain, the hottest and coldest points will tend toward the boundary over time. An equivalent statement is that a second Neumann eigenfunction of the Laplace-Beltrami operator has extrema only on the boundary. In 2021, Judge and Mondal proved the conjecture for Euclidean triangles by analyzing the behavior at the vertices and the zero-level sets of various derivatives of the eigenfunction. We will discuss their method of proof and demonstrate that it may be used with minor modifications to prove the conjecture for obtuse and right hyperbolic triangles.