I approach teaching mathematics from the viewpoint that basic mathematical literacy is part of a well-rounded higher education. I try to structure all of my courses in such a way as to emphasize broader mathematical ideas and themes, and to promote the development of general analytic and problem-solving skills. I feel that this can only be done effectively by promoting a fascination with and aesthetic appreciation for the material, through passionate and engaging instruction. At the same time, I am aware that the value of mathematics is different to different people, and depends largely on their other interests and career paths. Consequently, teaching demands versatility, with an independent study, say, requiring a separate approach from a large calculus lecture. Being mindful of my audience, and remembering that teaching must ultimately always be about the student, is one of the most important lessons I have learned as an instructor. Generally, I try to dissuade my students from the commonly held perception that the sole reason for studying mathematics is for its applications in science and industry. While applications are important, and I include their examples in my lectures whenever possible, not all students find them relevant. An arguably more important function of mathematics, and one that strongly influences my teaching philosophy, is that even in its simplest forms it is a unique tool for disciplining one’s mind, making everyday reasoning deeper and sharper. For this reason, I stress the importance of good mathematical writing and, in upper level courses, of proofs. I find that many of my students have a far better background in language arts than in mathematics, and are surprised to learn that their skills in the former can be useful in the latter. But being able to cleanly write out a computation, or to organize and present a complex idea, appeals precisely to these abilities. It is my hope that seeing mathematics as an elegant tool for communicating, rather than as an arcane set of rules for manipulating symbols, gives my students a better appreciation of mathematics as a liberal art. My views on teaching, and abilities as a teacher, continue to evolve. Ultimately, I try to keep an open mind, and remember that there will always be much for me to learn, both about mathematics and about mathematics teaching. In my own education, I was lucky to have had a number of very gifted and inspirational teachers, and I often think back to them and their methods in trying to shape my own. As an educator, I am motivated by a desire to similarly inspire my students. By sharing my own enthusiasm for the subject, I hope my students get a sense of the beauty of mathematics, and have a good time. I teach the gamut of usual undergraduate mathematics courses, such as calculus and linear algebra. In addition to these, I teach specialized undergraduate and graduate courses in the fields of mathematical logic and set theory.

Mathematical logic, specifically the areas of computability theory, reverse mathematics, and computable combinatorics. Mathematics today benefits from having “firm foundations”, by which we usually mean a system of axioms sufficient to prove the various theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid's geometry. Reverse mathematics is an area of mathematical logic that offers a modern approach to this kind of question, by classifying mathematical theorems according to their logical strength. This offers a deeper insight into the fundamental ideas and methods needed to prove a given theorem. More precisely, reverse mathematics provides a framework in which to compare and contrast results from disparate areas of mathematics, which helps elucidate the underpinnings of various branches of the mathematical sciences, and thereby leads to a better understanding of mathematics and its applications. A striking fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into one of a small number of categories. But for some very important and fundamental theorems this is not the case. My research focuses on theorems of this “irregular” type, including Ramsey's theorem, various equivalents of the axiom of choice, and principles arising from certain problems in cognitive science. In my current research program, I work to achieve a greater understanding of the complexities of these "irregular" theorems, to find new examples of such theorems from previously unexplored areas of mathematics, and to apply the reverse mathematics analysis to questions from outside of mathematics. This will be facilitated by the application of methods from computability theory and proof theory, and by the addition of ideas from various collaborations across a number of areas of pure and applied mathematics, as well as interactions with members of the multidisciplinary University of Connecticut logic group.

Since my earliest days in graduate school, I have been heavily involved in professional service. Before I graduated, I organized a number of meetings, including a major international workshop in mathematical logic. I refereed my first paper in my second year, and became an active reviewer for MathSciNet in my fourth. Since then, I have continued to serve on organizing and program committees, and to dutifully accept referee assignments and guest editorships. As I have benefited from the service of others, be it in the form of conferences I attended, or valuable feedback I received on a submitted paper, I strive to give the same quality of service back. This I regard not just as a duty, but also as a valuable way to get better acquainted with my colleagues and their work, and thus to stay better engaged in my field. In this way, I view service as a way to become a better member of my professional community.