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In this first lecture of a three-part series, Oliver Tse from Eindhoven University of Technology explores gradient structures that connect classical and quantum systems. Recorded at IPAM's Non-commutative Optimal Transport Tutorials at UCLA on March 12, 2025, discover how gradient flows provide a powerful framework for understanding dynamical systems through variational principles. The lecture begins with classical gradient structures and their applications in dissipative systems, explaining connections to transport costs and thermodynamics before transitioning to quantum analogs. Learn how these mathematical structures bridge perspectives between classical and quantum dynamics, offering insights into entropy dissipation and non-commutative transport. This 1-hour and 24-minute presentation serves as an introduction to the unified mathematical framework that connects concepts from thermodynamics, optimal transport, and quantum mechanics.
