Geometry & Topology

There are many striking analogies between two areas of geometry: minimal surface theory (in particular min-max theory) and spectral geometry (in particular nodal geometry). These include asymptotic Weyl laws, multiplicity-one theorems, density results, and more. In this talk, I will discuss one approach to a rather philosophical question: What is the source of these similarities? Is there a deeper theory or principle that explains them? In particular, I will explain a counterexample to the asymptotic optimality conjecture of Marques and Neves, and how a less rigid notion of nodal sweepouts nevertheless recovers Gromov's volume spectrum. The talk is non-expert–friendly and assumes no background knowledge.