TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

"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 will be completed with "Advanced topics in Analysis F" in the next quarter.

The students will become familiar with Beltrami equations, quasifoncormal mappings, and complex dynamics, and be ready for learning further applications.

Student learning outcomes

At the end of this course, students are expected to:
-- Understand the existence of isothermal coordinates
-- understand the geometric meanings of the Beltrami equation and quasiconformal mappings.
-- understand the solution of the Beltrami equation when its solution is quasiconformal.
-- be familiar with some basics of complex dynamics.

Keywords

Beltrami coefficient, Beltrami equation, quasiconformal mappings,
complex dynamics, Julia set, Fatou set

Competencies that will be developed

Class flow

This is a standard lecture course. Homework will be assigned occasionally

Course schedule/Required learning

Textbook(s)

None required

Reference books, course materials, etc.

Ahlfors, "Lectures on Quasifoncormal mappings", AMS
Carleson-Gamelin, "Complex Dynamics", Springer
Beardon, "Complex Dynamics", Springer

Assessment criteria and methods

assignments 100%.

Related courses

• MTH.C502 ： Advanced topics in Analysis F

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required