Amaranta Membrillo Solis (Queen Mary University) – Inverse spectral problems for orbifolds via the Hodge–Laplace operator

Orbifolds extend the concept of manifolds by allowing singularities that arise in a controlled way from group actions. They naturally appear in many geometric and physical settings, for example, as quotient spaces of symmetries, moduli spaces with isotropy, and as local models of singular spaces in mathematical physics. A fundamental problem in the spectral theory of orbifolds is whether the spectrum of a differential operator uniquely determines the underlying geometric or topological structure. This raises two natural questions:

(1) Can spectral data distinguish orbifolds with singularities from smooth manifolds?

(2) What geometric and topological features of the singular set can be recovered from spectral data?

Using heat invariants of the spectra of the Hodge–Laplace operator, we address these questions and examine how spectral information encodes the singular structure of orbifolds. This talk is based on joint work with Katie Gittins, Carolyn Gordon, Juan Pablo Rossetti, Mary Sandoval, and Liz Stanhope.