DMS Topology Seminar

Speaker: Brian Freidin (Auburn University)

Title: Eigenvalues and Morse theory for networks in spheres

Abstract: A minimal surface is a submanifold whose area does not change to first order under compactly supported variations. We will discuss the singular, one-dimensional analogue of this phenomenon - graphs embedded in a sphere. After describing the conditions for minimality, we will discuss second variations. The Morse index roughly counts the dimension of the space of length-decreasing variations of an embedding, while the nullity counts those variations that do not change length to second order. The nullity of a minimal embedding is related to an eigenvalue problem that appears in other applications, including heat and wave equations, and the local structure of harmonic maps between singular spaces.