Goals: - To analyze Möbius transformations with more than one fixed point in the extended complex plane - To continue with the classification of Möbius transformations begun in the previous lecture by defining the notions of loxodromic, elliptic and hyperbolic Möbius transformations using the values of the square of the trace of the transformation - To characterize geometrically the loxodromic, elliptic and hyperbolic Möbius transformations by showing that they can be conjugated by suitable Möbius transformations to multiplication by a complex number - To show that the elliptic Möbius transformations are precisely those that are conjugate to a rotation about the origin - To show that the hyperbolic Möbius transformations are precisely those that are conjugate to a real scaling Topics: Parabolic, elliptic, hyperbolic and loxodromic Möbius transformations, fixed point of a Möbius transformation, square of the trace of a Möbius transformation, translation, conjugation by a Möbius transformation, special linear group, projective special linear 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.