My latest research interests are given below. Students are encouraged to contact me to work on these projects with me.
Fractal Tree Topology: How do trees grow and how does the growth of trees relate to the canopy structure of forests? Interestingly, answers to these questions involve the study of fractal and relate to the growth of lung tissue. My projects in this area look at three dimensional tree structures and how to compute the structure of the fractals associated with trees. Computer programs help us simulate these tree structures and allow us to compute topological information.
Complex Dynamical Systems: I have several projects that look at how simple systems that change over time can lead to quite interesting behavior. In particular, we study the chaotic nature of these systems and realize how fractals play a role. Specific functional systems I study are the families of polynomial functions, sine functions, cosine functions, and exponential functions. We use computers to simulate the changing systems and to guide our mathematics.
Square-roots of Matrices: Understanding when matrices have square-roots leads to interesting investigations involving fractal pictures. Currently, I am investigating the square-roots of two by two matrices.