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. Topics: 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
The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.