Quantitative Topology and Geometric Inequalities

Explore a mathematical lecture that delves into quantitative interpretations of existence theorems in Geometric Calculus of Variations, originally proven through Algebraic Topology methods. Examine key theorems including Fet and Lyusternik's work on periodic geodesics in closed Riemannian manifolds, Serre's theorem on infinite geodesics between point pairs, and Zhou's recent findings on geodesic chords in complete manifolds. Learn how these quantitative versions yield geometric inequalities that connect minimal object sizes to spatial properties like volume, diameter, and curvature in the ambient space.