My research area is model theory, a subfield of mathematical logic. A broad perspective is that model theorists are interested in the underlying structure behind the number systems that mathematicians work with unquestioningly. In particular, the focus is often on definable sets - asking the question, "if I fix a set, a mathematical language (functions, relations, constants), and a set of rules for creating formulas in this language, what sorts of things are definable?"

A rough picture of where I fit in. Not to scale.

Example: In the natural numbers, allowing only multiplication, one can define the primes by a single formula (a formula which states "I am divisible only by myself and the identity, and I am not the identity."). But in the rational numbers, this same formula does not define the primes - in fact because the rationals are a field, every number is divisible by every other number save zero. As such the solution set to this formula over the rationals would be empty.

Specifically I am interested in the model theory of ordered structures (the real numbers, say, as opposed to the complex numbers), and even more specifically, my work has mostly been on weakly o-minimal structures.

A sparse timeline of the results leading to my work, from general to specific
(click on the faces for more information)

For a more long-form introduction, you can find a general overview of my research on the web here; also available here as a PDF. My latest slides are up on the talks and papers page.