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This lecture presents Aaron Palmer's research on "Non-Commutative Optimal Control and Viscosity Solutions" delivered at IPAM's Dynamics of Density Operators Workshop at UCLA. Explore joint work with Gangbo, Jekel, and Nam developing theory for optimal control problems with non-commutative variables inspired by dynamic random matrix models. Discover how classical descriptions transform into non-commutative analogues as matrix dimensions increase, creating mathematical challenges connected to free probability theory and quantum statistical mechanics. Learn about their approach using infinite-dimensional Hamilton-Jacobi-Bellman equations on non-commutative law spaces and their development of viscosity solutions adapted to non-commutative settings. Understand how this framework handles both classical "common noise" and non-commutative "free individual noise" randomness. The 50-minute presentation summarizes their progress, highlights unique challenges in non-commutative settings, and discusses open questions their research illuminates.
