The principal actors of complex analysis and geometry are holomorphic functions. Nonetheless, in the course of the 20th century another type of functions, called plurisubharmonic, rose to a most prominent supporting role. These functions are complex analogs (and at the same time, generalizations) of convex functions. They are relevant, because in many situations properties of plurisubharmonic functions have direct implications on properties of holomorphic functions.

Berndtsson's theorem, on a basic level, says that certain operations involving holomorphic and plurisubharmonic functions result in plurisubharmonic functions. After introducing the necessary notions, in this colloquium I will formulate an instance of the theorem, and discuss an application.

Weight representations are representations that decompose as direct sums of weight spaces. Examples of weight representations include highest weight representations, in particular, finite-dimensional representations. In this talk we will discuss the recent progress in the study of categories of weight representations over complex finite-dimensional Lie algebras and Lie algebras of vector fields. The talk is based on joint works with V. Serganova.

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