Course Outline - Mat 70400 Spring
2010
Course meetings, Tues
-Thurs 10:00 AM - 11:30 Room TBA
Texts:
Complex Analysis by L. Ahlfors , McGraw Hill
Hyperbolic Geometry
from a local viewpoint, L. Keen, N. Lakic, Cambridge.
Other classical texts you may want to consult are: Conformal Invariants by L. Ahlfors McGraw-Hill, Conformal Mapping by Nehari, McGraw-Hill, Complex Analysis by John Conway,
Springer, Modern Analysis by
Whittaker and Watson, Dover
Instructor:
Prof. Keen. Office Room 4302 Phone 212 817
8558 or 718 960 8867 email linda.keen@lehman.cuny.edu Office
hours: By appointment
Outline
I plan to
cover a selection of topics from among those listed below. I will assume that everyone in the
course has had a standard first semester complex analysis course and will rely
on material from such a course.
Contents of the course to include:
Harmonic
Functions,
Conformal mapping of simply connected domains, Schwarz-Christoffel maps,
Conformal mapping of multiply connected domains, Canonical domains
Elliptic
Functions, Basic definitions of Riemann surfaces, some hyperbolic geometry and Fuchsian groups
The first three weeks will cover Elliptic Function Theory
In the next two weeks , Prof. Jun Hu will cover Bloch's theorem, the little Picard theorem, Schottky's theorem, and the Montel-Caratheodory theorem and the Big Picard Theorem
In the next two weeks we will cover hyperbolic
geometry in the disk and in plane domains.
In the following week Prof Hu
will go over an old qualifying exam and Prof Basmajian
will talk about Mobius transformations.
After spring break we will cover the following
topics:
Schwarz-Pick theorems for the disk and arbitrary
domains
Distortion theorems
Conformal invaraiants and Extremal Length
Subharmonic Functions
Perron Method
Dirichlet's Problem
Harmonic measures
Green's functions
Introduction to Riemann surfaces
Homework assignments will appear on this page
approximately every other week. Students are strongly advised to work on all
the homework problems to make sure they are keeping pace with the class.
Homework is accepted
until the end of the term.
Students are encouraged to work together on homework, but each should
write out the solutions individually and indicate his/her collaborators.
The final grade will be based on the final exam
and the homework grades.
The final
exam will be the comprehensive exam, graded with respect to topics covered in
the class periods.
Homework
Assignment 2
Homework
Assignment 3
Homework
Assignment 4 (from Prof. Hu's classes)
Student Learning
Outcomes: After completing the second semester of this course students will be
able to:
· Carefully state and be able to apply the
major definitions and theorems of complex analysis.
· Understand the definition and major
theorems in the theory of elliptic functions, conformal mappings and harmonic
functions.
· Be able to understand and apply standard
theorems about transcendental
functions.
· Understand what constitutes a valid proof
of results in complex analysis and create such proofs.
· Be able to write mathematics in a precise,
effective, and understandable way.