Course Outline - Mat 70400 - Spring 2014


Course meetings, Tues -Thurs 2:30 PM - 4:00 PM pm  Room 6417



Texts: Complex Analysis by L. Ahlfors, McGraw Hill.
Conformal Invariants by L. Ahlfors, AMS Chelsea.
Hyperbolic Geometry from a local viewpoint, L. Keen, N. Lakic, Cambridge Other texts


Instructor:  Prof. Keen.  Office Room 4208.01  Phone 212 817 8531 or   email    Office hours:  By appointment




This is the second semester and will be a continuation of the course in Fall 2013. Assignments will be from the above texts or as indicated below.

Contents of the course will include:


Normal Families, Equicontinuity, Arzela's Theorem,  The Riemann Mapping Theorem, Dirichlet's Problem Explicit Conformal Maps

Elliptic Functions, General Theory of Periodic functions, Weierstrass Theory Linear

An introduction to hyperbolic geometry, Uniformization

Conformal Invariants, Extremal Length


Distortion Theorems


If there is time we will talk about quasi-conformal mappings and the basics of complex dynamics 


Homework assignments will appear on this page approximately every week. Students are strongly advised to work on all the homework problems to make sure they are keeping pace with the class.


The final grade will be based only on the homework grades.


The learning goals for this full year course are to master the basic ideas and tools of complex function theory: the ability to understand basic properties of analytic functions, Cauchy's theorem, Schwarz's Lemma, conformal mapping properties, integration theory for analytic functions, and basic theory of harmonic functions, entire functions, the Riemann mapping theorem. In addition students should have a working knowledge of many of the standard topics in Function Theory such as Periodic functions, Distortion Theorems, basic hyperbolic geometry.



Class Topic

Reading and Assignment

Jan 28 Equicontinuity and Normal Families,Ahlfors, Chap 5, sec 4
Jan 30 Class Cancelled Chap 5, sec 4, 4.5/1,2 due 2/1 1
Feb 4 Normal Families Cont'd. Riemann Mapping Theorem Chap 5, sec 4, 4.5/1,2 due 2/1 1
Feb 6 Riemann Mapping Theorem, Reflection Principle no assignment
Feb 11 Mappings of Polygons Chap 6, sec 2, 2.2/1,3 due 2/18
Feb 13 Rectangles and Triangles, Harmonic functions Chap 6, sec 2, 2.3/1,3 due 2/20
Feb 18 Harnack's principle, Dirichlet Problem Chap 6, sec 2, 3 2.3/1,3 due 2/20
Feb 20 Rectangles and Triangles, Harmonic functions Chap 6, sec 3, 4 sec 4.1/1,2 due 3/4
Feb 22 No Class Monday classes
Feb 25 Dirichet Problem, Canonical Domains Chap. 6, sec 5.1/1,2 Due Mar 11
Feb 27 Green's Functions, Periodic Functions Chap 6, Sec 5.2/1,2 Due Mar 13
Mar 4 Elliptic Functions
Mar 6 Elliptic Functions, cont'd.
Mar 11 Elliptic Functions, cont'd.
Mar 13 Elliptic Functions, cont'd. Chap 7, sec 3.21,2 Due Mar 18
Mar 18 Elliptic Functions, cont'd. Chap 7, sec 3.3/1,2,3,4,5,6,7 Due Mar 25
Mar 20 Elliptic Functions, cont'd.
Mar 25 Analytic continuation.
Mar 27 Riemann surfaces, flows
Apr 1 Riemann surfaces.
Apr 3 Riemann surfaces, cont'd. Exercise Sheet due Apr 24
Apr 8 Univalent functions
Apr 10 Univalent functions, cont'd. Hyperbolic geometry
Apr 15,17,22 Spring Break
Apr 24 Hyperbolic Geometry in the disk continued
Apr 29 Hyperbolic geometry in the upper half plane cont'd Exercise Sheet due Due May 8
May 1 Overview of covering maps and groups
May 6 Discontinuous Groups
May 8 hyperbolic geometry on general domains
May 13 No class
May 15 Extremal length