Department of Mathematics

Indian Institute of Science

Bangalore 560 012

 

SEMINAR

 

Speaker

:

 Ms. Eliza Philip
Affiliation : IISc, Bangalore.

Subject Area

:

Mathematics

 

Venue

:

Department of Mathematics, Lecture Hall I

 

Time

:

11.00 a.m.-12.00 p.m.

 

Date  

:

May23, 2012 (Wednesday)

Title

:

"Function Theory on non-compact Riemann surfaces"
Abstract

:

Given a domain D in the complex plane and a compact subset K, Runge's theorem provides conditions on K which guarantee that a given function that is holomorphic in some neighbourhood of K can be approximated on K by a holomorphic function on D. We look at an analogous theorem on non-compact Riemann surfaces, i.e., Runge's approximation theorem, stated and proved by Malgrange. We revisit Malgrange's proof of the theorem, invoking a very basic result in distribution theory: Weyl's lemma. We look at two main applications of Runge's theorem. Firstly, every open Riemann surface is Stein and secondly the triviality of holomorphic vector bundles on non-compact Riemann surfaces. Next, we look at the Gunning-Narasimhan theorem which states that every open (connected) Riemann surface can be immersed into $\mathbb{C}$. We discuss the proof of this theorem as well, which depends on Runge's theorem too. Finally we contrast the compact case to the non-compact case, by showing that every compact Riemann surface can be embedded into a large enough complex projective space.