Complex Monge-Ampère Operators and Functional Hermitian Intrinsic Volumes

Explore a geometric construction of Monge-Ampère-type operators defined on the space of finite convex functions in this 24-minute lecture. Delve into three distinct families of equivariant operators for convex functions on Cn, examining their integrability, continuity properties, and behavior under subspace restrictions. Investigate how these properties are reflected in valuations constructed from these operators, proposed as functional versions of Hermitian intrinsic volumes. Discover how these functionals provide a complete description of U(n)-invariant, continuous, and dually epi-translation invariant valuations on convex functions, drawing parallels to functional intrinsic volumes introduced by Colesanti, Ludwig, and Mussnig.