Goals: - To see that a Kleinian subgroup of Möbius transformations is a discrete subspace of the space of all Möbius transformations and also that such a subgroup is either finite or countable as a set - To define a subgroup of Möbius transformations to be Fuchsian if it maps a half-plane or a disc onto itself - To see that a discrete Fuchsian subgroup is Kleinian. For example, the unimodular group is thus Kleinian - To conclude using the results of the previous lecture that the quotient of the upper half-plane by the unimodular group is a Riemann surface Topics: Schwarz's Lemma, Riemann Mapping Theorem, properly discontinuous action, Kleinian subgroup of Möbius transformations, region of discontinuity of a subgroup of Möbius transformations, upper half-plane, unimodular group, projective special linear group, discrete subgroup of Möbius transformations, Fuchsian subgroup of Möbius transformations, holomorphic automorphisms, extended plane, stabilizer (or) isotropy subgroup, orbit map, second countable metric space, space of matrices, space of invertible matrices, space of determinant one matrices
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.