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This lecture presents Wilfrid Gangbo's research on "Viscosity solutions in non-commutative variables" delivered at IPAM's Dynamics of Density Operators Workshop at UCLA. Explore the development of stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in non-commutative variables, where inputs are tuples of self-adjoint operators from a tracial von Neumann algebra rather than real vectors. Discover how free semicircular Brownian motion replaces individual noise from mean field games, describing the large-n limit of Brownian motion on self-adjoint matrices. Learn about the introduction of classical common noise from mean field games into the non-commutative setting, allowing problems to combine both classical and non-commutative randomness. Understand how, under certain convexity assumptions, the value of optimal control problems in the non-commutative setting describes the large-n limit of control problems on tuples of self-adjoint matrices. This 53-minute talk, recorded on April 30, 2025, is based on collaborative work with D. Jekel, K. Nam, and A. Palmer.
