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Math 428: Topics in complex analysis

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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) =
q^{1/24} ** P**_{n
}
(1-q^{n})

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

hassett@math.rice.edu

http://www.math.rice.edu/~hassett