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: * To characterize discrete subgroups of the additive group of complex numbers and to use this characterization to classify Riemann surfaces whose universal covering is the complex plane

* To see how the twisting of the universal covering space by an automorphism (of the universal covering space) leads to the identification of the fundamental group (of the base of the covering) with a conjugate of the deck transformation group of the original covering

* To show that the natural Riemann surface structures, on the quotient of the complex plane by the group of translations by integer multiples of a fixed nonzero complex number does not depend on that complex number; in other words that all such Riemann surfaces are isomorphic

Keywords: Upper-triangular matrix, complex plane, universal covering, deck transformation, abelian fundamental group, additive group of translations, module, submodule, discrete submodule, discrete subgroup