Spectral Multiplicity and Nodal Domains of Torus-Invariant Metrics

Explore a 52-minute seminar on spectral geometry focusing on spectral multiplicity and nodal domains of torus-invariant metrics. Delve into Samuel Lin's presentation, part of the "Spectral Geometry in the clouds" series, which examines Uhlenbeck's classical result on simple Laplace spectra for generic Riemannian metrics. Investigate how symmetries in manifolds affect spectral simplicity, particularly when compact Lie groups act as isometries. Learn about the conjecture from quantum mechanics and atomic physics regarding eigenspaces as irreducible representations for generic G-invariant metrics. Discover Lin's proof of this conjecture for torus actions and his findings on nodal sets of real-valued, non-invariant eigenfunctions. Compare these results with previous work on nodal domains for surfaces with ergodic geodesic flows, highlighting the stark contrast in the number of nodal domains. Gain insights into this joint project with Donato Cianci, Chris Judge, and Craig Sutton, presented at the Centre de recherches mathématiques (CRM).