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
- Elliptic Curves and Elliptic Functions
- 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
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