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 interpret a cylinder as a suitable quotient of the complex plane;

– To use the above interpretation to give a Riemann surface structure on a cylinder and

to raise the question as to how many such non-isomorphic structures exist.

Keywords for Lecture 4:

Translation by a complex number, equivalence relation, equivalence class, set modulo an equivalence relation, glueing edges of a strip, inverse image of an equivalence class, quotient topology, quotient map, open map, homeomorphism, Moebius transformation, group action on a set, orbits of an action, set modulo (action of) a group