Recorded 30 April 2025. Wilfrid Gangbo of the University of California, Los Angeles, presents "Viscosity solutions in non-commutative variables" at IPAM's Dynamics of Density Operators Workshop. Abstract: Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equa-tions in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean eld games is replaced by a free semicircular Brownian motion, which describes the large-n limit of Brownian motion on the space of self-adjoint matrices. We introduce a classi-cal common noise from mean eld games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness. Under certain convexity assumptions, we show that the value of the optimal control problems in the non-commutative setting describes the large-n limit of control problems on tuples of self-adjoint matrices. (This talk is based on works in collaboration with D. Jekel, K. Nam and A. Palmer). Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-dynamics-of-density-operators/