R509, Cosmology Building, NTU
Speaker:
ChingJui Lai (National Cheng Kung University)
Organizers:
ChingJui Lai (National Cheng Kung University)
1. Background
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.
2. 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, RiemannRoch Theorem, AbelJacobi 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 AbelJacobi map

Theory of divisors

Finiteness theorem and RiemannRoch formula

Applications of RiemannRoch formula

Residue map and Serre duality

Existence of meromorphic functions (from calculus of variations, harmonic functions, to constructions of meromorphic fucntions)

A proof of RiemannRoch formula

Uniformisaztion Theorem
3. Evaluation

Attendance and Discussion 20%

Homework 30%

Final project 50%
4. Reference:

Rick Miranda, Algebraic Curves and Riemann Surfaces, (Graduate Studies in Msrhematics, Volume 5)

Simon Donaldson, Riemann Surfaces, 2011.
5. Credit: 3
Contact:
murphyyu@ncts.ntu.edu.tw