Abstract: In 1906, F. Hartogs discovered the existence of domains in $\mathbb{C}^n$ for which every holomorphic function can be extended to a larger domain. Domains that do not admit this extension phenomenon satisfy a complex type of convexity, known as pseudoconvexity. This type of convexity can be viewed as convexity "with respect to holomorphic functions", as opposed to classical convexity which is convexity "with respect to linear functions". $$ $$ In this talk, we will motivate and define pseudoconvexity. We will also compare and contrast its many equivalent formulations with that of classical convexity. We will also introduce a class of "pseudoconvex" manifolds known as Stein manifolds and discuss their many properties. Notions of convexity with respect to other classes of functions will also be discussed.

Talk 6 of the learning seminar on Pontryagin rings in toric topology.