Riemann Surfaces :: Math 9302 :: Summer 2014

Course information

Instructor:

Rasul Shafikov, MC 112, (all email inquiries will be answered within 48 hours).
Office hours: TBA, in MC 112.

Course Coordinates:

Course meets Tuesdays and Thursdays 10 AM - 12 PM in MC 108. The first class is on Tuesday May 6, and the last class is on July 10. There will be two weeks with no classes: May 19-23, and June 23 - 27. There is no class on July 1. Instead we will meet on June 30, 2 - 4 PM in MC 108. OWL Login

Prerequisites:

A solid undergraduate course in Complex Analysis. Some familiarity with basic topology, and with covering spaces.

Evaluation

Homework:

There will be 4 homework assignments that should be submitted for marking. Tentative dates for submitting assignments:

1. Homework assignment 1 due May 27;
2. Homework assignment 2 due June 5,
3. Homework assignment 3 due June 19,
4. Homework assignment 4 due July 3,

Presentation:

At the end of the course every registered student will give a 1/2 hour presentation on a topic prearranged with the instructor.

Evaluation:

Homework = 60%, Presentation = 40%.

Further Details

Course Content

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely "useful" part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry and diverse topics in mathematical physics. (S. Donaldson).
The course is intended as a rigorous introductory course to Riemann surfaces. The material can be divided into four major parts.

1. Abstract Riemann surfaces (basic definitions, examples, algebraic curves, quotients, function theory on Riemann surfaces) Short summary of lecture is here.
2. Topology of Riemann surfaces (vector fields and differential forms, Stokes theorem, De Rham cohomology, surgery on Riemann surfaces, type of a Riemann surface). Summary of lecture is here.
3. The Uniformization theorem (harmonic and subharmonic functions, Dirichlet problem, Green's function, covering spaces) Material for this portion of lectures can be found in Gamelin's "Complex Analysis".
4. The Riemann-Roch theorem (...)

Topics for Presentations

1. Resolution of singularities (normalization) of algebraic curves (Huang) Tuesday, July 8
2. Harnack's principle for bounded families of harmonic functions (Yau) Tuesday, July 8
3. Complex structure on smooth orientable surfaces (Hazratpour) Thursday, July 10
4. Approximation on Riemann surfaces (Sharifi) Tuesday, July 8
5. Harmonic measures (Ussher) Thursday, July 10
6. Necessity of elliptic functions for the classification of complex tori. (Alluhaibi), Thursday, July 10.

References

Material will be taked from several texts, the list is below. Some textbooks will be on reserve in Taylor library.

1. O. Forster. Lectures on Riemann Surfaces. Springer, GTM 81.
2. S. Donaldson. Riemann Surfaces. Oxford Graduate Texts in Mathematics, 22.
3. T. Gamelin. Complex Analysis. Springer, UTM. 2001.

Senate Regulations on Scholastic Offences

Please note the following points, which are required to be stated in this outline by the Senate regulations.

Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following Web site: http://www.uwo.ca/univsec/handbook/