In Summer 2003 the Mathematics Department of the Pennsylvania State University hosted a Research Experiences for Undergraduates site. Support for the site was provided by the National Science Foundation VIGRE grant. Thirteen participants were selected from qualified applicants.

The program ran for 7 weeks from **June 23 to August 8, 2003**. It combined learning with research and included:

- Two
**two-week mini-courses***Elliptic Curves and Elliptic Functions*

Instructor: Yuri Zarhin, Professor of Mathematics*Introduction to Complex Dynamics*

Instructor: Bryna Kra, Assistant Professor of Mathematics

**Research projects**- Paul Baginski and Elena Fuchs,
*Explicit formulas for the modular equation* - Jonathan Barry and Timothy Trudell,
*Isogenic but non-isomorphic elliptic curves over finite fields* - Walter Chen and Mike Willis,
*Properties of the iterates of the Weierstrass P-function* - Yi Fang Chen, Vivek Srikrishnan,
*Minimal non-composite rectangle exchange transformation* - Nathan Collins, Jeff Ginn, Sara Salko,
*Drawing closed geodesics* - David Jordan and Rhiannon Schayer,
*Rational points in the Cantor middle thirds set, Group structure on Cantor rationals*

- Paul Baginski and Elena Fuchs,

## Courses

#### ELLIPTIC CURVES AND ELLIPTIC FUNCTIONS

INSTRUCTOR: YURI ZARHIN

MAIN TOPICS include:

- Plane algebraic curves
- A rational parametrization of quadrics and singular cubics
- Elliptic functions; Divisors and principal divisors; The Weierstrass P-function; The field of elliptic functions
- Elliptic curves; A parametrization of elliptic curves by elliptic functions; Addition theorems for the Weierstrass P-function and its derivative
- A characterization of the divisors of elliptic functions
- The projective line, projective plane and infinite points; The group law on elliptic curves

#### INTRODUCATION TO COMPLEX DYNAMICS

INSTRUCTOR: BRYNA KRA

MAIN TOPICS include:

- Iteration of functions, fixed points, periodic points, attracting and repelling fixed points
- Dynamics of the quadratic family
- Transitivity, sensitive dependence to initial conditions
- Chaos, Julia and Fatou Sets
- Mandelbrot Set