# Course

## M659 - Topology and Complex Analysis

Course No:
M659
Credit:
4
Approval:
2014
PG-Core
Syllabus:

Topology: Topological spaces, Continuous maps between topological spaces, product topology, Quotient spaces, Connectedness, Compactness,Winding Numbers of Closed Curves, Brouwer Fixed Point Theorem (statement only), Borsuk-Ulam Theorem (Statement Only). [20 lectures]Complex Analysis: Complex line integrals, Goursat’s theorem; Local existence of primitives and Cauchy’s theorem in a disc , Cauchy’s integral formula , Applications of Cauchy’s integral, Singularities and their classifications, zeros, poles and residue theorem. Applications of residue theorem. Argument principle and applications. Maximum Modulus principle, Schwarz lemma. Biholomorphic between between complex plane, Disc to itself, Statement of Riemann Mapping theorem. [30 lectures]

Reference Books:
1. Armstrong, Basic Topology, Springer, 1983
2. Munkres, Topology, Pearson Education, 2005.
3. Greene and Krantz, Function Theory of One Complex Variable, gsm 40, University Press, 2006
4. Stein and Shakarchi, Complex Analysis (Princeton Lectures in Analysis, No. 2), Princeton University Press, 2003.
5. Gamelin, Complex Analysis (Undergraduate Texts in Mathematics), Springer, 2003.
6. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.