A Fixed Point Theorem for Isometries of Metric Spaces

Explore a groundbreaking fixed point theorem for isometric maps of metric spaces in this 48-minute lecture from the Thematic Programme on "Geometry beyond Riemann: Curvature and Rigidity" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the concept of weak convex bicombing, which can be interpreted as Busemann nonpositive curvature for a specific class of geodesics. Discover how this theorem applies to various spaces, including Banach spaces, CAT(0)-spaces, injective metric spaces, and the space of positive operators with Thompson's metric. Learn about the innovative approach of using metric functionals as an extension of Busemann functions to provide fixed points even when classical methods fail. Examine practical applications of this theorem, including a new mean ergodic theorem generalizing von Neumann's theorem for Hilbert spaces, and the discovery of a non-trivial invariant metric functional on the space of positive operators for any invertible bounded linear operator of a Hilbert space.