### Pólya's eigenvalue conjecture: some recent advances

Analysis and Geometry Seminar

27th April 2023, 3:30 pm – 4:30 pm

Fry Building, Room 2.04

Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. The conjecture is known to be true for domains which tile the Euclidean space, however it remains largely open in full generality. In the talk we will explain the motivation behind this conjecture and discuss some recent advances, notably, the proof of Pólya’s conjecture for the disk. The talk is based on a joint work with Nikolay Filonov, Michael Levitin and David Sher.

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