Geometric Spectral Invariants and Isospectrality

Licentiate thesis, 2025

This thesis investigates how geometric features of domains are reflected in the spectrum of the Laplacian, a central theme in spectral geometry. In Chapter 1, we introduce the subject and explain classical results and questions, such as Weyl’s law, Milnor's 16-dimensional pair of flat tori, and Kac’s question “Can one hear the shape of a drum?” In Chapter 2, we study integrable polygonal domains and obtain explicit expressions for associated spectral invariants, including the spectral zeta function and the heat trace. We show that, for this class of polygons, the length of the shortest closed geodesic appears in the remainder of the heat trace expansion. We also analyze the convergence of heat trace coefficients under geometric limits between convex polygons and smooth domains. Chapter 3 presents the first ever example of a 6-dimensional triplet of isospectral but non-isometric flat tori, and we explain how it relates to previously known results. In Chapter 4, we study the short-time heat trace expansion of convex polygons with Neumann boundary conditions and obtain an explicit remainder estimate using locality principles, extending results previously known only in the Dirichlet case. Finally, in Chapter 5, we conclude by summarizing the main contributions of the thesis and outlining directions for future research.