Gradient structures provide a powerful framework for understanding the evolution of dynamical systems by linking them to variational principles. This lecture series explores the role of gradient flows in classical and quantum settings, illustrating how they unify concepts from thermodynamics, optimal transport, and quantum mechanics. We begin with classical gradient structures, discussing their applications in dissipative systems and their connections to transport costs and thermodynamics. Moving to the quantum realm, we examine quantum analogs of gradient flows, exploring their role in open quantum systems, quantum entropy dissipation, and non-commutative transport. By bridging these perspectives, we hope to highlight the emerging mathematical structures that offer new insights into both classical and quantum dynamics.
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