"The Beltrami equation" is a partial differential equation whose homeomorphic solutions realize a given deformation of (1 dimensional) complex structure in terms of the "Beltrami coefficient". When the norm of "the Beltrami coefficient" is strictly less than one, its homeomorphic solution is called a "quasiconformal mapping", which is an essential tool for the theories of Teichmullar space, Kleinian groups, and complex dynamics. In this course we will start with several solutions Beltrami equations, then we will learn some basics on complex dynamics and significant applications of quasiconformal mappings. This course is a succession of "Advanced Topics in Analysis E" in the previous quarter.
The students will learn some remarkable applications of the Beltrami equations and quasifoncormal mappings in the theory of complex dynamics.
At the end of this course, students are expected to:
-- be familiar with some basics of complex dynamics.
-- understand the principle of quasiconformal perturbation and quasiconformal surgery
Beltrami coefficient, Beltrami Equation,quasiconformal mappings, complex dynamics, Julia sets, Fatou sets
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. Homework will be assigned occasionally
Course schedule | Required learning | |
---|---|---|
Class 1 | Classification of stable domains in complex dynamics | Details will be provided in class. |
Class 2 | Fatou-Shishikura inequality 1 | |
Class 3 | Fatou-Shishikura inequality 2 | |
Class 4 | Polynomial-like mappings and traightening | |
Class 5 | Sullivan's no wandering domain theorem 1 | |
Class 6 | Sullivan's no wandering domain theorem 1 | |
Class 7 | Degenerate Beltrami equations of David class | |
Class 8 | Parabolic surgery, an application of degenerate Beltrami equations of David class |
None required
Ahlfors, "Lectures on Quasifoncormal mappings"
Carleson-Gamelin, "Complex Dynamics", Springer
Beardon, "Complex Dynamics", Springer
assignments 100%.
None required