Course
Outline  Mat 70400  Spring 2013
Course meetings, Tues
Thurs 4:15 PM  5:45 PM pm Room 6417
Instructor:
Prof. Keen. Office Room 4208. Phone 212
817 8531 or
Email: LKeen@gc.cuny.edu Office hours: Tues, Thurs 24 pm and by
appointment
Texts:
Complex Analysis by L. Ahlfors, McGraw Hill (some material from the 3rd edition). Functions
of One Complex Variable I, John B. Conway, Springer,
Hyperbolic Geometry from a Local Viewpoint by L.Keen and N. Lakic, Cambridge 2007
Cambridge Univ. Press, 2007
Other texts you may want to consult are: Conformal Invariants, Topics in Geometric Function Theory by L. Ahlfors, McGrawHill 1973, Theory of Functions by Caratheodory, Chelsea, The five small volumes on Theory of Functions by K. Knopp, dover reprint, A course in Modern Analysis by Whittaker and Watson.
Outline:
This course is a continuation of the fall semester course. We will cover most of the remaining material in Ahlfors: normal families, Riemann Mapping Theorem, Multiplevalued functions and elliptic functions. Other topics this semester will include: extremal length, distortion theorems, basic hyperbolic geometry and an introduction to Riemann surfaces.
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. Assignments in Ahlfors are given by page numbers in the hard cover third edition. The Chapter and Section are given for those with a different version.
The final grade will be based on the homework grades. Students may take the qualifying exam to improve their grade.
Class and Homework Assignments


Date 
Class
Topic 
Reading
and Assignment 
Jan 29  Normal Families  Ahlfors p. 219 (Chap.5 Sec 4.5) 1,3,4 Due Feb 14 
Jan 31  Riemann Mapping Thm  Ahlfors p. 224 1,2 Due Feb 14 
Feb 5  Mapping polygons  Guest lecturer Thomas  Ahlfors p. 230 (Chap 6. Sec 2.2) 2,3,5 Due Feb 19 
Feb 7  Harmonic Functions redux  
Feb 12  No Class  Lincoln's Birthday  
Feb 14  Subharmonic Functions Dirichlet Problem  p. 239 (Chap 6 Sec 4.1) 2,3,4 Due Feb 26 
Feb 19  mapping multiply connected domains, Green's functions  Conway p.277,1,2,4 Due Mar 7 
Feb 21  Elliptic Functions I  
Feb 26  Elliptic Functions II  Chap 7, Sec 3.2, 1,2 Due Mar 14 
Mar 5  Elliptic Functions III note class will end at 5:15 today  Chap 7, Sec 3.3, 1,2,3,4 Due Mar 21 
Mar 7  Elliptic Functions IV  
Mar 12  Analytic Continuation  
Mar 14  Analytic Continuation  Ahlfors Chap 7, sec 1.3/2, Due Apr 4 Note Change 
Mar 19  Monodromy, branch points and the Picard theorem  
Mar 21  Algebraic Functions and Linear Differential Equations  
Mar 26  Spring Break  
Mar 28  Spring Break  
Apr 2  Spring Break  
Apr 4  Hyperbolic Geometry  
Apr 9  Discontinuous Groups  KeenLakic Chap 5 and 6 
Apr 11  Discontinuous Groups Chap 6  
Apr 16  Discontinuous Groups, Dirichlet domains  HW Sheet 
Apr 18  Discontinuous Groups, Poincare's theorem  
Apr 23  Hyperbolic Geometry Chap 7 , hyperbolic metric on plane domains  
Apr 25  Hyperbolic Geometry, properties of hyperbolic metrics  
Apr 30  Hyperbolic Geometry Chap 7, Schwarz Pick Lemma  
May 2  Extremal Length, Ahlfors Conformal Mapping  
May 7  Distortion Theorems  
May 9  Extremal Domains  
May 14  Extremal Domains and Elliptic functions  
May 21  Qualifying Exam 2pm to 5pm 