Goals of the Lecture:- To look at the set of all possible Riemann surface structures on a cylinder and the need for a method to distinguish between them- To explain the motivation for the use of the Theory of Covering Spaces to distinguish Riemann surface structures- To motivate the notion of a covering map by examples- To get introduced to the fact (called General Uniformisation) that any Riemann surface is the quotient (via a covering map) of a suitable simply connected Riemann surface- To understand the idea of the Fundamental group and where it fits into our discussionKeywords for Lecture 6:Cylinder, punctured plane, punctured unit disc, annulus, Riemann's Theorem on removable singularities, covering map, covering space, pathwise connected, locally pathwise connected, admissible neighbourhood or admissible open set, universal covering space or simply connected covering space, fundamental group, uniformisation of a general Riemann surface
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.