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Explore geometric inequalities and quantitative topology in this mathematics lecture that delves into existence theorems in Geometric Calculus of Variations proven through Algebraic Topology methods. Learn about A. Fet and L. Lyusternik's theorem on periodic geodesics in closed Riemannian manifolds, J. P. Serre's theorem on infinite geodesics between point pairs, and X. Zhou's recent findings on geodesic chords in complete manifolds. Discover how quantitative versions of these theorems generate geometric inequalities that connect minimal object sizes to spatial properties like volume, diameter, and curvature in the ambient space.
