Introduction to Systolic Geometry - Lecture 1

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 two significant Gromov results: the upper bound proof for high genus surface systoles through Kodani's methodology, and Nabutovsky's recent breakthrough providing improved constants for aspherical manifold systole bounds. Gain insights into Berger's 1960s conjecture regarding aspherical manifold generalizations and understand how these mathematical developments have shaped our understanding of geometric topology.