Abstracts for Reading group on spectral geometry

The aim of this talk is to introduce classical methods of dealing with eigenvalues on manifolds. We will begin with variational characterization via  minimax principles, then we will talk about topological restraints arising form Courant's nodal domain theorem. From there we turn towards isoperimetric problems introducing Cheeger constant and proving Cheeger's inequality - establishing a lower bound for smallest eigenvalue. If time allows we will show Buser's upper bound for $\lambda_1$ and provide some examples.

This talk will begin with an introduction to the length spectrum of a compact hyperbolic surface and a derivation of the Length Trace Formula. We will then use this formula to prove Huber’s Theorem, establishing the equivalence between the length spectrum and the eigenvalue spectrum of the Laplacian. Finally, we will examine the trace of the heat kernel and its relationship to the eigenvalues, specifically focusing on the proof of the pretrace formula.

We will present Sunada's construction, which allows to build examples of isospectral Riemannian manifolds that are in general non-isometric. We will apply this construction to one specific case in order to show the existence of isospectral but not isometric pairs of hyperbolic surfaces. Time permitting, we will discuss some related results.

This talk will introduce expander families and the combinatorial Laplacian, emphasizing the role of spectral gaps in measuring expansion. It will then discuss how group-theoretic constructions provide natural families of graphs with uniform spectral gaps.