Goals: To show that the various annuli with inner radii in the real open unit interval and with outer radius unity are all non-isomorphic as Riemann surfaces Topics: Upper half-plane, universal covering, fundamental group, deck transformation group, Möbius transformations, real special linear group, real projective (special) linear group, simply connected, biholomorphic map, holomorphic isomorphism, infinite cyclic group, parabolic Möbius transformation, hyperbolic Möbius transformation, fixed point, extended plane, abelian fundamental group, commuting Möbius transformations, commuting deck transformations, punctured unit disc, annulus, unique lifting property
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.