Geometric Analysis and Topology Seminar

---

Local Marked Spectral Determination for ℤ₂-symmetric planar billiards close to the disk (for now)

Speaker: Jacopo De Simoi, University of Toronto

Location: Warren Weaver Hall 512

Date: Friday, December 6, 2024, 11 a.m.

Synopsis:

Given a planar domain with sufficiently regular boundary, one
can study periodic orbits of the associated billiard problem. Periodic
orbits possess a rich and intricate structure: it is then natural to ask
how much information about the domain is encoded in the set of lengths
of such orbits. The quantum analog of this question is the celebrated
Laplace inverse problem, or “Can one hear the shape of a drum?"

We prove Marked Dynamical Spectral Determination among ℤ₂-symmetric
smooth convex domains close to the disk: if any two such domains have
the same Marked Length Spectrum, they must necessarily be isometric
domains.  This substantially improves the deformational
result obtained in a prior work with Kaloshin and Wei.