IISc Bengaluru, IASc Asociate: 2020 (Mathematics)
Session 2D Lectures by Fellows/Associates
The role of topology in certain analytic problems
There are several analytic phenomena that set apart the multivariate theory of complex-analytic functions from its univariate counterpart. Some of these give rise to notions of convexity which form the subject of many open problems in several complex variables. In this talk, we will give an overview of this subject in the special, but natural, setting of real submanifolds in complex Euclidean spaces. Here, some relevant factors are: the topology of the submanifold, the (partial) complex structure inherited by the submanifold from the ambient space, and the nature of the singularities of this inherited structure. Questions of convexity (and of the associated hulls) can be considered both when all these factors are rigid, or, when the embedding of the submanifold is allowed some flexibility. We will discuss some recent results in both these flavors, while emphasizing the modern aspects of this theory.