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Math 428: Topics in Complex Analysis

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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