Research activities

I am a dynamicist with specific research interests in topological dynamical systems, both pure and applied. Below I give brief details about my own work and projects with collaborators. The central theme of my research in dynamical systems is computer experiment as a means of supporting and proving conjectures.

Fractal billiards, flat surfaces, IETs

C. Johnson and I are currently investigating the flow on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, …

Mathematical Andreev billiards

Mathematical Andreev reflection is retro-reflection with parity. By this, we mean that when a pointmass intersects a portion of a billiard table not obeying the standard Law of Reflection, the pointmass returns upon the incoming path. In addition to this, the ball is assigned a parity and such parity changes with each intersection with said …

Mathematical sociology

In collaboration with R. E. Niemeyer and under the auspices of the AMIS lab (Applications of Mathematics and Integrated Science), we examine the effect of particular parameter changes in a system of ordinary differential equations modeling the interaction between people of a nation-state, the amount of internal strife and the amount of resources within the system. …

Numerical experiments and simulations

Additional work has focused on computational modeling of number-theoretic problems so as to gain insight into the behavior of particular systems. Joint work with D. Bradley, A. Khalil and E. Ossanna has focused on finding a rigorous connection between a matrix representation of Pascal’s triangle modulo a square-free integer and a planar representation of such a matrix, with the immediate …