Math 428: Topics in complex analysis

MWF 10:00AM, Herman Brown 453

Assignments and lecture notes

This course will focus on one topic typically covered in a second course in complex analysis: Elliptic function theory. Elliptic functions-as developed by Jacobi, Weierstrass, Eisenstein, Dedekind, and others-are one of the crowning achievements of 19th century mathematics and are widely applied in physics and engineering. Their study is the natural continuation of the analysis of polynomial, exponential, and trigonometric functions of a complex variable. In the 20th century, the analysis of the beautiful transformation properties of elliptic functions developed into the theory of elliptic curves and modular forms, a central topic of algebraic geometry and number theory. Recently, elliptic functions have played an important role in the 21st century mathematics inspired by theoretical physics. For example, the Dedekind eta function

h(q)   =   q1/24 Pn (1-qn)
occurs in work of S.T. Yau and E. Zaslow on BPS states (a.k.a., the enumeration of nodal rational curves on K3 surfaces).

Here are some specific topics we may explore:

complex tori and doubly periodic functions;
s,z,h,q functions and the Weierstrass P-function;
modular functions and the Picard theorem;
applications to number theory (representing integers as sums of squares and quadratic forms).

References: I have ordered the following textbook from the bookstore:

K. Chandrasekharan, Elliptic functions, Springer-Verlag, Berlin, 1985.
It should arrive the first week of class. In preparing my lectures, I will draw on books on elliptic functions by Lang, Weil, and others.

Assessment: Problems will be assigned regularly. Each problem is due the second Monday after it is assigned (i.e., the first homework is due September 9). This will account for 60% of the final grade. There will be a take-home, closed-book final exam which will account for the remaining 40% of the grade.

Contact information:
Brendan Hassett
Herman Brown 422
(713) 348-5261