I work in low-dimensional topology. The objects I study are 3-manifolds and the knots that live in them, as well as ways to construct or classify these objects such as Dehn surgery and homology cobordism. Many of the tools I use come from Heegaard Floer theory and related areas (I studied with Zoltan Szabo), and some of my projects also involve a direct study of Heegaard Floer theory itself, its invariants, and their properties. My approach to mathematics may be characterized as calculational or combinatorial; I am very interested in the ways in which we calculate the properties of an object and the typical behavior of these invariants for random objects as well as the kinds of things that can go wrong in extreme cases. I think this is a good way to learn about the objects.
A toolkit of many of my mathematical calculations is available. Please experiment and let me know what you learn!