### MA7014 Riemann Surfaces and Algebraic Curves

#### Course Details

**Description:**

To introduce the students to the geometry of Riemann surfaces, both from analytic as well as algebraic viewpoints, highlighting the rich interplay between the topological, analytical and algebraic aspects of compact Riemann surfaces and their moduli.

**CourseContent:**

Topology of Riemann Surfaces : Definitions and examples of Riemann surfaces, Euler-Poincare and Riemann-Hurwitz formulae, Galois Theory of Coverings, Uniformisation, Kleinian, Fuchsian and Elementary Groups.

Analytic Theory of Riemann Surfaces : Holomorphic differentials, Integration, Weyl's Lemma, Meromorphic functions.

Algebraic Aspects of Riemann Surfaces : Compact Riemann Surfaces as Algebraic Curves, Riemann-Roch Theorem, Abel's Theorem, Jacobian Variety of a Riemann Surface, Jacobi Inversion

#### Course References:

TextBooks:

1. H. M. Farkas, I. Kra, Riemann Surfaces, Graduate Texts in Mathematics 71, Springer-Verlag, 1980.

2. R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics vol.5, American Mathematical Society, 1995.

References:

1. O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer-Verlag, 1981.

2. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1997.

3. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves - I,Grundlehren der mat.Wissen.267, Springer-Verlag, 1985