MATH GR6175

Basics of the theory of Riemann Surfaces

Knowledge: Theta functions, Weierstrass p-functions, heat kernel approach to Riemann-Roch Perspective: Analytic

Weierstrass functions, theta functions, the complex torus, the general theory of Riemann surfaces, Riemann-Roch theorem

Crash course in complex variables Constructing holomorphic and meromorphic differential forms via: theta functions, weierstrass p function, basic PDE methods through the lens of the glued cut plane torus Basic Riemann surfaces, Riemann roch theorem, tiny bit of cohomology theory

Good for learning the basics of the theory of Riemann surfaces

Only take this if you are comfortable with differential forms 1 Columbia University: Arts & Sciences Fall 2021 Course: MATHGR6175_001_2021_3-COMPLEXANALYSIS/RIEMANNSURF : MATHGR6175_001_2021_3 - COMPLEX ANALYSIS/RIEMANN SURF Instructor: Duong Phong

Basic Complex Analysis, Riemann Surfaces, Tori, Holomorphic Line Bundles, Covariant Derivative, Curvature, Chern Class

Basic concepts about Riemann surfaces with genus g, Riemann Roch theorem, some intro to index theorem, Abel theorem on Riemann surfaces, and some knowledge about comlex manifolds and PDE's defined on sections of the bundles.

Basic examples of Riemann surfaces, function theory on torus, line bundles, Riemann-Roch, compact Riemann surfaces with genus g...

Began with local theory of holomorphic and meromorphic functions on the complex plane, including proving equivalent definitions of holomorphicity and the residue theorem. Studied riemann surfaces as the proper setting of analytic continuation of certain functions next. Then, studied the complex torus, elliptic integrals, Weierstrass theory, theta functions, and the d-bar equation. Studied Abel map and Jacobi inversion theorem. Presented the heat kernel proof of Riemann-Roch (although we didn't prove the functional analysis) and ended by discussing higher-genus surfaces and their theta functions.