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:    Office hours:  Tues, Thurs 2-4 pm and by appointment



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.




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, Multiple-valued 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

Keen-Lakic Chap 2 and 5



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 Keen-Lakic 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