Goals of the Lecture: - Every good topological space possesses a unique simply connected covering space called the Universal covering space - The fundamental group of the topological space shows up as a subgroup of automorphisms of its universal covering space - The universal covering map expresses the target space as the quotient of the universal covering space of the target, by the fundamental group of the target - A covering map can be used to transport Riemann surface structures from source to target and vice-versa, thus making it into a holomorphic covering map - Any Riemann surface is the quotient of the complex plane, or the upper half-plane, or the Riemann sphere by a suitable group of Möbius transformations isomorphic to the fundamental group of the Riemann surface - The study of any Riemann surface boils down to studying suitable subgroups of Möbius transformations Topics: Covering map, covering space, admissible open set or admissible neighborhood, simply connected covering or universal covering, local homeomorphism, Riemann surface structure inherited by a topological covering of a Riemann surface, uniformisation, fundamental group, Möbius transformation
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.