Course No.： S070101ZJ008
Course Category：Professional Basic Course
Prerequisites： basic theory of one complex variable, basic topology.
Aims & Requirements：
Complex analysis I is a basic course of graduate students of all majors in mathematics, it also an optional course for physical students. The course is aimed to lay part of the foundation for doing research work in related areas. After learning this course, students are hoped to be able to understand basic ideas and concepts of modern complex analysis and know well the history of developments of this subject.
Chapter 1. Riemannian surfaces
Definitions and examples, covering surfaces, fundamental groups, deck transformation groups.
Chapter 2. The uniformization theorem
Green’s functions, harmonic measures and the maximum principle, classification of open Riemannian surfaces, proof of the uniformization theorem.
Chapter 3. Riemann-Roch’s theorem
The De Rham cohomology groups, holomorphic differentials, meromorphic differentials and bilinear relations, divisors and Riemann-Roch’s theorem, proof of Riemann-Roch’s theorem, Weierstrass’s gap theorem, Abel’s theorem and its corollaries.
Chapter 4. Quasiconformal mappings
The geometric definition, differentiable K-q.c. mappings, distributional derivatives, analytic properties of q.c. mappings, the existence theorem and its consequences, distortion theorems.
Chapter 5. Boundary correspondence
Quasi-circles and quasiconformal reflections, quasi-symmetry and the Beurling-Ahlfors’s extension.
Chapter 6. Extremal quasiconformal mappings
Main inequalities, necessary and sufficient conditions for being extremal
q.c. mappings, Teichmuller mappings.
Chapter 7. Teichmuller spaces
The universal Teichmuller space, Bers’ embedding, holomorphic motions.
1. Lars V. Ahlfors, Lectures on quasiconformal mappings, 1966/1987/2006 (3 Editions).
2. Olli Lehto, Univalent Functions and Teichmuller Spaces, Springer-Verlag, 1987
3. John Hubbard, Teichmuller Theory and Applications to Geometry, Topology, and Dynamics, vol. 1, Teichmuller Theory, Matrix Editions, 2006.
Author：Gui-zhen Cui（Academy of Mathematics and Systems Science）