Duration: 23:38

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I will present a geometric construction of certain Monge-Amp`ere-type operators defined on the space of finite convex functions. For convex functions on Cn, this naturally gives rise to three distinct families of equivariant operators with rather different integrability and continuity properties. In addition, they differ substantially in their behavior under restrictions to subspaces. We will discuss how these properties are reflected in the valuations constructed from these operators, which we propose as
functional versions of the Hermitian intrinsic volumes, which where originally introduced by Bernig and Fu. In fact, these functionals provide a complete description of the space of U(n)-invariant, continuous, and dually epi-translation invariant valuations on convex functions. In this sense, these valuations are the Hermitian analog of the functional intrinsic volumes introduced by Colesanti, Ludwig, and Mussnig.