2021-2022 honors thesis

Gerardo GOnzalez

I had the pleasure of advising Gerardo Gonzalez on his honors thesis, Relating the Numerical Range of the Composition Operator of an Anti-Diagonal Matrix to the Convex Hull of Ellipses, successfully defended in Spring 2022. This work was an extension of Gerardo's research in the summer of 2020 at an REU (Research Experience for Undergraduates) at the University of Michigan-Dearborn, with Dr. John Clifford. The composition operator of a 2x2 anti-diagonal matrix turns out to have a convex hull of ellipses as its numerical range. Gerardo noticed that the parameters for the anti-diagonal matrix that produced a family of ellipses that were inside the first (thus producing a numerical range that was an ellipse) when plotted on a 3D graph produced a curious image, that I thought looked a little like a quasar. He wanted to determine the exact parameters and equation that produced this “quasar” shape. He skillfully condensed and explained graduate level mathematics, produced original research that answered his question in a complete and satisfactory way, and wrote dozens of pages of Mathematica code. Gerardo is now pursuing his Ph.D. in Physics at Texas A&M University.

2009-2010 Honors Theses

Sean Watson and Tommy Rogers

I had the pleasure of directing two honors theses for two amazing students, Sean Watson and Tommy Rogers. Sean’s honors thesis, “On the Convergence of Infinite Series and Products”, explored the idea of infinite series in a number of different settings, including infinite dimensional spaces, Banach spaces, and Lie groups. He explored a new version of the Dvoretsky-Hanani theorem for abelian Lie groups, and also for the non-abelian Heisenberg group. Tommy’s honors thesis titled, “Super Resolution Image Construction”, combined two techniques for reconstructing a higher resolution image from a collection of lower resolution images of the same scene. This involved applying a number of areas of mathematics, including Fourier analysis, matrix theory, and group theory. 

Tommy and Sean both presented their papers at the SU Research and Creative Works Symposium as well as in Portland at MathFest, the national meeting of the Mathematical Association of America. Sean received a Pi Mu Epsilon Excellent Speaker Award for his presentation.