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 2-4 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, 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
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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 | Keen-Lakic Chap 2 and 5|
| 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 | |