Explore the mathematical foundations of optimal transport in this 54-minute lecture by Alessio Figalli from ETH Zurich, recorded at UCLA's Institute for Pure & Applied Mathematics (IPAM) during the Statistical and Numerical Methods for Non-commutative Optimal Transport Workshop. Delve into the critical topic of regularity theory for optimal maps, examining its theoretical underpinnings and significant applications in the field. This presentation offers valuable insights for researchers and students interested in advanced mathematical concepts related to optimal transport theory. For additional resources and information about the workshop series, visit IPAM's website at https://www.ipam.ucla.edu/programs/workshops/workshop-iii-statistical-and-numerical-methods-for-non-commutative-optimal-transport/.