Introduction to Systolic Geometry - Lecture 4

Explore systolic geometry in this one-hour lecture that delves into the fundamental concept of systole in Riemannian manifolds - the length of the shortest non-contractible loop - and its relationship to manifold volume. Learn about Loewner's pioneering 1949 findings for the torus case and examine Berger's subsequent questions about aspherical manifolds. Gain detailed insights into two significant Gromov results: the upper bound proof for high genus surface systoles through Kodani's demonstration, and Nabutovsky's recent breakthrough providing improved constants for aspherical manifold systole bounds.