Goals of the Lecture: - To get an idea of the classification of Riemann surfaces that can be arrived at based on the fundamental group, using the theory of covering spaces - To get introduced to the notions of: moduli problem, moduli space, number of moduli, fine and coarse classification, and to write these down for simple Riemann surfaces Topics: Biholomorphic map or isomorphism of Riemann surfaces, classification of Riemann surfaces, universal covering of a Riemann surface, abelian fundamental group, complex plane, unit disc, upper half-plane, punctured plane, punctured unit disc, cylinder, complex torus, annulus, Riemann sphere, g-torus, coarse classification, fine classification, moduli problem, moduli theory, moduli space, number of moduli
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.