MATH 5121: Topics in Complex Function Theory
Description: Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others. With a change of content this course may be repeatable to a maximum of twelve credits.
Prerequisites: MATH 5120.
MATH 5121 - Section 1: Riemann Surfaces
Description: The Riemann surface was first studied by Riemann as a geometric configuration of multi-valued functions. Locally a Riemann surface looks like the complex plane, but the global geometry and topology is very different. Studying complex analysis on Riemann surfaces lead to surprising connections with topology and manifold theory. The first part of the course will discuss holomorphic functions and give an informal discussion on classification of surfaces, in order to set-up the interplay between complex analysis and topology for later topics. The second part will discuss Riemann surfaces and holomorphic maps with plenty examples, elliptic functions and integrals, and applications of Euler characteristic. The third part will cover the fundamental results on Riemann-Roch Theorem and the Uniformization Theorem. The course is for beginning graduate students with some background in general topology and complex analysis. The course will also cover the necessary materials on the relevant differential topology and algebraic topology in the case of surfaces.
Sections: Fall 2017 on Storrs Campus
|07274||5121||001||Lecture||TuTh 02:00:00 PM-03:15:00 PM||MONT227||Huang, Lan-Hsuan|