Riemann Surfaces

Code: MA647 | L-T-P-C: 3-0-0-6

Prerequisites: MA 547 (Complex Analysis)

Course Content/ Syllabus: Riemann surfaces: definitions, examples, basic properties, degree and genus; Analytic continuation: Riemann surface of an analytic germ; Differential forms on Riemann surfaces: Weyl's lemma and the Hodge decomposition, Examples of meromorphic functions and differentials; Homology bases and holomorphic differentials, Periods and bilinear relations, Divisors, Riemann-Roch and the theorems of Abel and Jacobi; Uniformization

References:

1. Wilhelm Schlag, A course in Complex Analysis and Riemann Surfaces, Graduate Studies in Mathematics: Volume 154, American Mathematical Society, 2014.
2. Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, Vol. 22, Oxford University Press, Oxford, 2011.
3. Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Series No. 5, American Mathematical Society, 1995.
4. Dr. Heinrich Durè, Elements of the theory of functions of a complex variable with especial reference to the methods of Riemann, Authorized translation from the 4th German Edition (1893) by George Egbert Fisher and Isaac J. Schwatt, Norwood