**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.

**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.

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

- J. E. Fornaess, F. Forstnerič and E. F. Wold, Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan.
- N. Leveneberg, Approximation in ${\mathbb C}$
^{$n$}.

- 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.