Symplectic Weyl Laws

Abstract: Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics and C^0 symplectic geometry. For example, it plays a key role in the proof of the smooth closing lemma for three-dimensional Reeb flows and area preserving surface diffeomorphisms, and in the proof of the simplicity conjecture. ECH and PFH are highly sophisticated theories whose construction in particular relies on Seiberg-Witten theory. I will explain how one can use much more elementary methods (no Floer or gauge theory) to define spectral invariants satisfying an analogous Weyl law with similar applications.