Refined asymptotics for the ECH spectrum

In previous work, Hutchings, Ramos and I studied the embedded contact homology (ECH) spectrum for any closed three-manifold with a contact form, and proved a “volume identity” showing that the leading order asymptotics recover the contact volume. I will explain recent joint work with Savale that sharpens this asymptotic formula by better understanding the subleading term. I will then discuss some conjectures about what an optimal bound on the subleading asymptotics might be: for example, for certain generic domains in R^4, Hutchings has conjectured that the subleading asymptotics recover a contact “Ruelle invariant”.