Convexity Seminar

Remarks on convex domains of maximal symplectic size

Location: Warren Weaver Hall 1302

Date: Tuesday, March 24, 2026, 11 a.m.

Synopsis:

Symplectic capacities are invariants coming from various techniques in symplectic geometry that measure the size of symplectic manifolds. Although convexity is not preserved under symplectomorphisms, convex domains nevertheless display special behavior with respect to these capacities. Viterbo’s volume--capacity conjecture (2000) suggested that, among convex domains of a fixed volume, the Euclidean ball maximizes every symplectic capacity. A special case of this conjecture is equivalent to the famous Mahler's conjecture. By capturing a deep connection between convex and symplectic geometries, Viterbo’s conjecture has become a central problem in the study of symplectic capacities and has inspired extensive research. For example, one result in this direction shows that smooth symplectic Zoll domains—a class defined by a special dynamical property—are local maximizers.

In this talk, I will describe a counterexample to Viterbo’s conjecture, obtained jointly with Yaron Ostrover, and discuss some of the questions that arise from it. In particular, the counterexample shows that a capacity maximizer cannot be both smooth and strictly convex. This leads to the problem of identifying nonsmooth dynamical features that detect local maximizers. I will propose a dynamical extension of the Zoll property to the nonsmooth case and discuss its connection with certain topological conditions.