An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals of the Lecture:

To develop a suitable definition of a structure of a Riemann

Surface on a 2-dimensional surface that will allow us to carry out

Complex Analysis (i.e., study of holomorphic (or) analytic functions)

on the given surface.

Keywords:

Complex plane, open set, analytic (or) holomorphic function, Cauchy-Riemann

equations, complex differentiable, convergent power series, Taylor expansion,

Taylor coefficients, open map, biholomorphic map (or) holomorphic isomorphism,

homeomorphism (or) topological isomorphism, complex coordinate chart,

compatibility of charts, transition functions, Riemann surface structure