# Jerusalem Mathematics Colloquium

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Thursday, 27th December 2001, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Irwin Kra

(SUNY, Stony Brook)

Uniformizations and
*theta*-constant identities

### (Based on joint work with Hershel M. Farkas)

** Abstract: **
One variable theta functions are useful tools in complex analysis, number
theory and combinatorics. They provide examples of uniformizations of
Riemann surfaces that can serve as models for a more general theory.

After
motivating an approach to the study of one variable *theta*-functions
through problems arising from Kleinian groups and Ahlfors' finiteness
theorem, we will take a leisurely tour of an elliptic paradise and its
number theoretic and combinatorial regions, including a discussion of
multiplicative functions related to the Ramanujan *tau*-function. The
emphasis will be on function theoretic derivations of *theta*-constant
identities and their use to uniformize (compact) Riemann surfaces. If time
permits we will discuss products of *theta*-constants that result in
constant functions, identities **not** connected to Riemann surface
theory, and infinite product expansions of *theta*-constant
derivatives (generalizing the Jacobi derivative formula).

Coffee, Cookies at the faculty lounge at 3:30.

List of talks, 2001-02

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