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 lkeen@gc.cuny.edu Office hours: By
appointment
Outline
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
Distortion Theorems
If there is time we will talk about quasiconformal 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.


Date 
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 