My research area is the topology and geometry of 3-manifolds. I was attracted to it because of the richness it acquired from Thurston’s revolutionary work starting in the 1970s. His key insight was that many 3-manifolds admit homogeneous Riemannian metrics, and that one can study the topology of a 3-manifold via this geometry. This profusion of geometry has now been stunningly confirmed by Perelman’s proof of the Geometrization Conjecture. As a direct result, while my work has focused on what initially seem like purely topological problems, in fact I have used a broad range of techniques to attack them, including hyperbolic geometry, number theory, and algebraic geometry, as well as more obviously related areas such as combinatorial group theory and the theory of foliations. These connections to other fields have led me to collaborate with number theorists, theoretical physicists and computer scientists, and in my papers I've used both the Langlands Conjecture and the Classification of Finite Simple Groups, as well as such topological oddities as “random 3-manifolds”.Since the summer of 2007, I have been an Associate Professor here at the University of Illinois at Urbana-Champaign. Previously, I spent four years at Harvard and four years at Caltech after getting my PhD from the University of Chicago sometime back in the 20th century.
None till Fall 2014
Office: 378 Altgeld