'Approximation Theory in Complex Analysis' by Purvi Gupta - June 9, 2023 - Session 1

Being able to approximate functions from a given class by ’nice’ functions such as polynomials or rational functions is a powerful and time-tested tool in analysis. Along with the pioneering works of Weierstrass and Runge, approximation theory in (one variable) complex analysis is driven by well-known results due to Mergelyan, Vitushkin, Arakelian, and others. Not only are these results extremely useful in themselves, attempts to generalize them to broader settings have led to the development of new techniques and notions in mathematics. On the one hand, approximation theory in several complex variables borrows techniques from a wide range of topics such as functional analysis, potential theory, singular integral representation theory, and PDEs (specifically, the so-called dbar method). On the other hand, this theory has many applications, particularly to embedding problems, giving this subject a geometric flavour. In this course, we will first review the proofs of a selection of one-variable results, focusing on abstract (rather than constructive) techniques since these are more adaptable to higher dimensions. We will then discuss the obstructions in generalizing these results to several complex variables, as a consequence of which we will encounter different notions of convexity. At this stage, we will emphasize certain topological aspects of this theory. Towards the end of the course, we will mention some recent developments in this subject.