Research Interests and their Evolution:
In my early explorations of mathematics, my interests were focused on computational geometry, graph theory and probability, spurred on by several excellent teachers and mentors, in the person of both fellow students and faculty, and generally through immersion in the vibrant mathematics cultures of Stuyvesant High School and the Hampshire College Summer Studies in Mathematics. I considered myself a devotee of pure mathematics, drawn in by the promise of absolute rigor and led to read both philosophers and mathematicians in the positivist tradition, like Boole, Frege, and Russell, to understand the character of rigor and the varying notions of proof mathematicians have held.
More recently, my historical and philosophical reading turned to people like Wittgenstein, Karl Popper, and Thomas Kuhn, and I find my current (avocational) foundational interests leaning towards the influence of mathematical thought in the history and philosophy of science. As a teacher and an aspiring researcher, I am fascinated by accounts of how mathematicians really think, and how mathematical concepts really come to be: the real mathematics, done by humans and constrained by how our brains work, that is only later distilled into straightforward theorem-proof-corollary exposition in mathematics textbooks. I believe that detailed understanding of our mental processes and the history of mathematics is an invaluable aid and guide in the classroom and the laboratory both.
This shift was in large part occasioned by a desire to bring mathematical tools to bear on other, often non-quantitative fields that have held my attention and occupied my time again and again: evolutionary biology–particularly biomechanics, developmental mechanisms of evolutionary change, and the practical uses of sequence analysis; music–its deep history and how it is processed by both human and machine; and the workings of language, literature, and music on the brain. I hope to focus in mathematical biology, but my educational aims thus far have been to acquire tools from mathematics, physics and statistics that have wide applicability (as well as beauty of their own), and one of my greatest joys has been talking to students in other fields and using my mathematical skills to help their projects. These several fields of application may all profitably be pursued, while in the possession of large data sets, through methods in statistics, signal processing, mechanics and applied analysis, lending some coherence to my research interests to date.
Research Programs and Papers in Preparation:
In high school, I was the 4th place winner in the 2008 Intel Science Talent Search for a project in combinatorial geometry. I am currently revising and expanding this preprint, which I will soon post here.
In the summer of 2010, I participated in the mathematics REU at East Tennessee State University under the direction of Anant Godbole. I recommend this program enthusiastically! The mentoring, the location, and the mathematics made for a wonderful experience in my mathematical and personal life both.
While at ETSU, I worked on two joint projects, both presented at the Southeastern REU Mini-Symposium in July 2010 and the Joint Mathematics Meetings in January 2011.
With Dr. Godbole and Nicholas Triantafillou, a UofM student, I worked on omnimosaics, matrices and other structures containing all possible substructures over a certain set. The preprint is available on the arXiv here. This work is inspired by the work of Fan Chung and others on universal graphs.
While working on the omnimosaics project, I also met with elementary mathematics teachers who were working on a parallel problem to discuss how to present this research in the classroom. Those discussions were possible through an NSF GK-12 grant given to the ETSU mathematics department.
The other project was a collaboration with fellow REU student Chang-Mou Lim to study the efficient domination of infinite graphs endowed with additional symmetries and geometric structure, building on the extensive literature on graph domination number for finite graphs, especially tiling graphs. This work is still in preparation.
In the summer of 2011, I studied Mark Kac’s problem “Can You Hear the Shape of a Drum?” and its solution by Gordon, Webb and Wolpert, and later Conway, with the suppport of a Harvard College PRISE Fellowship.