Return to Mentoring activities

Student Projects

I encourage interested students to contact me about doing either an undergraduate honors thesis or an independent research project.  Students working with me will learn about fractal geometry, dynamical systems, mathematical billiards, flat surfaces and interval exchange transformations.  In addition to that, students will learn to program and understand the value of a computer algebra system in proving theorems about fractal billiards, fractal flat surfaces and fractal interval exchange transformations.

Below are some topics that can be modified or expanded upon to suit a student’s interest and abilities.

Simulating billiards on the Koch snowflake fractal billiard table

The Koch snowflake fractal billiard table is a fractal billiard table with nowhere differentiable boundary. A priori, reflection at any of the points is not well-defined. Building on previous works, you will be asked to simulate orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

Simulating billiards on the T-fractal billiard table

The T-fractal billiard table is a somewhat nicer fractal billiard table in that a fair amount of the boundary of the billiard table yields a well-defined tangent. You will be asked to simulate the orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

Broader subject of dynamics

For those more interested in the broader subject of dynamics, one can also compare and contrast the notions of ergodic, weak mixing and strong mixing. This project is best suited for an undergraduate student finished with advanced calculus and interested in learning about measure theory. In preparing the student for the senior thesis, one will first be asked to read and understand the proof of the Poincare Recurrence Theorem and the definitions involved.