Goals of the Lecture: - To interpret the torus as a suitable quotient of the complex plane; - To use the above interpretation to give a Riemann surface structure on a torus and to raise the question as to how many such non-isomorphic structures exist. Topics: Translation by a complex number, equivalence relation, equivalence class, set modulo an equivalence relation, glueing edges of a parallelogram, inverse image of an equivalence class, quotient topology, quotient map, open map, homeomorphism, Möbius transformation, group action on a set, orbits of an action, set modulo (action of) a group
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.