Speaker: Lorenzo Sarnataro (Toronto)

Title: Index, Intersections, and Multiplicity of Min-Max Geodesics

Abstract: The p-widths of a closed Riemannian surface are geometric invariants associated with the length functional. In a recent work, Chodosh and Mantoulidis showed that these invariants are realised as the weighted lengths of unions of closed immersed geodesics (possibly, with multiplicity). I will discuss joint work with Jared Marx-Kuo and Douglas Stryker, where we prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the p-width of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on S^2 for which the p-width is only achieved by p copies of a single closed geodesic.