Math 428: Topics in Complex Analysis

MWF 1:00-1:50, Herman Brown 427

Description: The main topic will be complex Riemann surfaces. Riemann surfaces are at the crossroads of complex analysis, algebraic topology, differential geometry, and algebraic geometry. This course will introduce them using the techniques of complex function theory, but we will develop the geometric and topological aspects as well. We will not assume any prior background in topology or geometry; anything we need will be developed as we go along.

Here are some specific topics we will explore:

Examples of Riemann surfaces

Holomorphic and meromorphic functions

Holomorphic maps of Riemann surfaces

Branched coverings and monodromy

Integration on Riemann surfaces

Divisors and maps to projective space

Riemann Roch Theorem and applications

Textbook: The main textbook should be available from the bookstore:

Algebraic Curves and Riemann Surfaces, Rick Miranda

American Mathematical Society

For general background from complex analysis, we refer to

Complex Analysis, Stein and Shakarchi

Princeton University Press

Prerequisites: A semester course in complex analysis

Assessment: Weekly problem sets (40%): Due each Monday, except for the first set which is due September 6. They should be turned in during class. Homework assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission.

Midterm exam (20%): A closed-book 80-minute take-home test, to be taken sometime between Friday, September 29 and Monday, October 2.

Final exam (40%): A closed-book three-hour take-home exam.

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities will need to also contact Disability Support Services in the Ley Student Center.

Contact information:
Brendan Hassett
Office: Herman Brown 402
Phone: (713) 348-5261
Email: hassett@math.rice.edu
Webpage: http://www.math.rice.edu/~hassett