Recorded 11 March 2025. Tryphon Georgiou of the University of California, Irvine, presents "Schrödinger Bridges: Old and New, Part 1 of 2" at IPAM's Non-commutative Optimal Transport Tutorials. Abstract:In 1931 Erwin Schrödinger published a paper with the title "Über die Umkehrung der Naturgesetze" (On the Reversal of the Laws of Nature), where he explored the time reversal of the law of a diffusion process and its implications when conditioning the law to satisfy specified marginals at two points in time. The law of the conditioned process, with time-marginals that interpolate the specified end-point marginals, came to be known as a Schrödinger bridge. Schrödinger's ideas linked a rather broad spectrum of concepts that, in modern language, include the relative entropy between probability laws, likelihood estimation, large deviations theory, stochastic optimization and Monge-Kantorovich optimal mass transport. The aim of these two lectures is to overview the mathematics and applications of SBs both in a classical as well as the non-commutative setting. Specifically, in part I, we will present snipets of the theory and applications of classical Schrödinger bridges, and in part II, we will develop and motivate non-commutative counterparts and their potential applications. Learn more online at: https://www.ipam.ucla.edu/programs/long-programs/non-commutative-optimal-transport/non-commutative-optimal-transport-tutorials/?tab=schedule