Approximation Theory

Lectures will be held online on MS Teams (at least for the first four weeks). The team link/code is available on the IISc intranet (see Department of Mathematics). If you are not from IISc, drop me an email to be added to the team.

Instructor. Purvi Gupta. The best way to contact me is via email (purvigupta(at)iisc(dot)ac(dot)in).

Course timings. Tuesdays & Thursdays, 10:00 am - 11:30 am.

Office hours. To consult me outside of class, drop me an email to set up an appointment.

Resources. All the material for this course will be posted on the MA 324 MS Teama page.

Brief description. Approximation-theoretic results play an important role in complex analysis, especially in the construction of holomorphic functions with desired properties. Thus, sets that admit good approximation-theoretic properties deserve special attention. Runge's approximation theorem, for instance, has led to such notions as polynomial convexity and Oka manifolds. The goal of this course is to familiarize the audience with some results and techniques in complex-analytic approximation theory.

Part I. After reviewing a selection of classical approximation and interpolation results (and proofs) on $\mathbb C$, we will move to the setting of (open) Riemann surfaces. We will see analogues of results such as the Mittag-Leffler theorem, the Weierstrass factorization theorem, the Runge approximation theorem, etc., on open Riemann surfaces. The techniques will be a combination of potential-theoretic tools, and the so-called $\overline{\partial}$-method, all of which will be discussed as and when needed. The end goal will be to see some geometric applications, including the proper holomorphic embeddability of open Riemann surfaces into $\mathbb{C}$$3$.

Reference for Part I. D. Varolin, Riemann Surfaces by Way of Complex Analytic Geometry. Graduate Studies in Mathematics, 125, American Mathematical Society, Providence, 2011.

Part II. The length and breadth of this part will be determined by how much time is available after the conclusion of Part I. After reviewing a few more results in $\mathbb C$ (not covered in Part I), we will discuss some notions that have arisen in the process of extending these results to higher dimensions. No prior knowledge of several complex variables will be assumed.

References for Part II. Material will be sourced from various places (which will be cited in due course).

To get a broad sense of the overall theme of this course, you can browse through the following two survey articles.

  • Hard prerequisite: MA 224 (Complex Analysis).
  • Soft prerequisites: some knowledge of potential theory in $\mathbb{C}$ (MA 226) and Riemann surfaces (MA 307) will be hepful. However, all the necessary background from these topics will be covered as and when needed.