Week 24.02.2025 – 02.03.2025

The existence of spectral asymptotics of Laplace or Schrödinger operators acting on Riemannian manifolds is a classical problem known for more than 100 years. It has been known for a long time that obstacles to the existence of spectral asymptotic expansions are periodic and looping trajectories of the geodesic flow. A conjecture formulated in 2016 stated that these trajectories are the only such obstacles. I will discuss the history of this problem and describe the resent progress: proving this conjecture in special cases, as well as constructing some counterexamples.