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 firstname.lastname@example.org 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
Elliptic Functions, General Theory of Periodic functions, Weierstrass Theory Linear
An introduction to hyperbolic geometry, Uniformization
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.
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|