Third Year Major Oral: The Scalar Curvature Problem via Dirac Operators: From Closed Manifolds to Manifolds with Boundary

Abstract: The interplay between positive scalar curvature and topology is a central theme in global analysis. One powerful tool for addressing these problems is the use of the Dirac operator and index theory, which are rooted in the classical Lichnerowicz Vanishing Theorem and the Atiyah-Singer Index Theorem. In this talk, I will first review these foundational results and introduce the method of using twisted Dirac operators to study the curvature properties of a Riemannian metric. I will then present how Llarull, inspired by the developments of Gromov and Lawson, adapted the technique to prove his rigidity theorem: any Riemannian metric on the (n)-sphere that dominates the standard round metric cannot have strictly greater scalar curvature. Finally, I will discuss the challenges and some recent developments in extending these quantitative rigidity results to manifolds with boundaries.

Faculty Advisor: Xiang Tang

Note: This talk is given in partial fulfillment of the Third Year Candidacy Requirement.