There are beautiful and unexpected connections between algebraic topology, number theory, and algebraic geometry, arising from the study of the configuration space of (not necessarily distinct) points on a variety. In particular, there is a relationship between the Dold-Thom theorem, the analytic class number formula, and the "motivic stabilization of symmetric powers" conjecture of Ravi Vakil and Melanie Matchett Wood. I'll discuss several ideas and open conjectures surrounding these connections, and describe the proof of one of these conjectures--a Hodge-theoretic obstruction to the stabilization of symmetric powers--in the case of curves and algebraic surfaces. Everything in the talk will be defined from scratch, and should be quite accessible.

In 1966, Marc Kac in his famous paper 'Can one hear the shape of a drum?' raised the following question: Is a bounded Euclidean domain determined up to isometries from the eigenvalues of the Euclidean Laplacian with either Dirichlet or Neumann boundary conditions? Physically, one motivation for this problem is identifying distant physical objects, such as stars or atoms, from the light or sound they emit.

The only domains which are known to be spectrally distinguishable from all other domains are balls. It is not even known whether or not ellipses are spectrally rigid, i.e. whether or not any continuous family of domains containing an ellipse and having the same spectrum as that ellipse is necessarily trivial.

In a joint work with Steve Zelditch we show that ellipses are infinitesimally spectrally rigid among smooth domains with the symmetries of the ellipse. Spectral rigidity of the ellipse has been expected for a long time and is a kind of model problem in inverse spectral theory. Ellipses are special since their billiard flows and maps are completely integrable. It was conjectured by G. D. Birkhoff that the ellipse is the only convex smooth plane domain with completely integrable billiards. Our results are somewhat analogous to the spectral rigidity of flat tori or the sphere in the Riemannian setting. The main step in the proof is the Hadamard variational formula for the wave trace. It is of independent interest and it might have applications to spectral rigidity beyond the setting of ellipses. The main advance over prior results is that the domains are allowed to be smooth rather than real analytic. Our proof also uses many techniques developed by Duistermaat-Guillemin and Guillemin-Melrose in closely related problems.

The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon.