Introduction to Riemann Surfaces
2021/9/15-2022/1/21 10:10-13:10 on Wednesdays
Rm 509, Chee-Chun Leung Cosmology Hall, National Taiwan University( Distance Learning)
Speaker:
Ching-Jui Lai 賴青瑞(National Cheng Kung University)
Organizer:
Ching-Jui Lai 賴青瑞(National Cheng Kung University)
Background & Purpose
The subject of compact Riemann surfaces or algebraic curves has its origin going back to the work of Riemann. Its development requires ideas from analysis, PDE, differential geometry, complex geometry, algebra, and topology e.t.c. This course serves as a introductory course to more general theories of complex manifold and higher dimensional algebraic geometry.
Outline
In this course, we will introduce the notion of Riemann surfaces, holomorphic functions, meromorphic functions, differential forms on Riemann surfaces, maps between Riemann surfaces, Riemann-Roch Theorem, Abel-Jacobi theorem. We might discuss the existence of meromorphic functions and the uniformization theorem if time allows.
Tentative plan:
Formal definition of a Riemann surface: From CP^1 to real/complex manifolds.
Examples of Riemann surfaces: quotient, graph, affine curves, projective spaces and projective curves
Holomorphic and meromorphic functions
Rational polynomials and theta functions
Holomorphic maps of Riemann surfaces
Hurwitz formula: proof and applications
Differential forms, Stoke Theorem, and Residue Theorem
Holomorphic one forms and Abel-Jacobi map
Theory of divisors
Finiteness theorem and Riemann-Roch formula
Applications of Riemann-Roch formula
Residue map and Serre duality
Existence of meromorphic functions (from calculus of variations, harmonic functions, to constructions of meromorphic fucntions)
A proof of Riemann-Roch formula
Uniformisaztion Theorem
Evaluation
Attendance and Discussion 20%
Homework 30%
Final project 50%
Reference:
1. Rick Miranda, Algebraic Curves and Riemann Surfaces, (Graduate Studies in Mathematics, Volume 5)
2. Simon Donaldson, Riemann Surfaces, 2011.