I am interested in simple problems in geometry, topology, and dynamical systems. Much of my research deals with the consequences of allowing a simple pattern or phenomenon to repeat forever. The kinds of patterns and constructions I study are idealized versions of what you would see in everyday life, such as a billiard ball bouncing around on a table, or the pattern of shapes on a turtle shell, or a beam of light reflecting in a series of curved mirrors.

In general, my work falls into the area of infinite groups, geometric topology, projective and hyperbolic geometry, and dynamics. Here are the 5 examples of projects I have been involved in during the course of my career:

1. infinite group theory and geometric rigidity: An infinite finitely generated group has associated to it a Cayley graph, a combinatorial object which captures a lot of the large-scale geometry of the group. One can ask whether different groups give rise to Cayley graphs which are, in some sense, roughly the same. I have studied this problem for a natural collection of groups which arise in geometry and number theory, namely the nonuniform lattices of Q rank one. These include fundamental groups of hypobolic knot complements, and Hilbert modular groups. I proved a sweeping classification theorem for these groups, showing essentially that even a fairly blind person could
recognize the group by looking at its Cayley graph.

2. Groups of symmetries of the complex hyperbolic plane: The complex hyperbolic plane is a four-dimensional curved symmetric space. I have studied infinite groups of symmetries acting on this space. My most recent result in this area is a general surgery theorem which tells how the associated 3-manifold at infinity changes during the deformation of a discrete group with nonempty domain of discontinuity. Some of this work was the subject of my address at the International Congress of Mathematicians in 2002. I have recently completed a research monograph, to be published in the Princeton Annals of Math series, which details my surgery theorem.

3. Dynamical systems based on geometric constructions: Here one starts with a simple geometric construction, such as barycentric subdivision or Pappus's theorem, and creates a dynamics system based on repeated applications of the construction. I investigate these constructions with the use of a computer and then try to prove theorems about what I observe. For instance, I proved that the iteration of barycentric subdivision of an N dimensional simplex produces a dense set of shapes for N less or equal to 4.

4. triangular billiards: Recently I proved that a triangular billiard table has a periodic billiard path, provided that all its angles are at most 100 degrees. I discovered this result using McBilliards, a graphical user interface that I co-wrote with Pat Hooper to study this problem. My result represents the first substantial progress on the 200-year-old Triangular Billiards Problem, which asks if every triangular shaped billiard table has a periodic billiard path.

5. Outer billiards is a simple dynamical system based on a convex planar shape. This system was introduced by B.H. Neumann in the 1950's. All along the central question in the subject was: Does there exist a shape for which the corresponding system has an
unbounded orbit. Recently I solved this problem by showing that outer billiards has an
unbounded orbit when defined relative to any irrational kite. A kite is a kite-shaped
quadrilateral.

1. Continuous NSF grant support since 1998:
Highly competitive, three-year grants from the Division of Mathematical Sciences of the National Science Foundation for $50,000-100,000 per year to cover summer salary, graduate student support, travel, and equipment. These grants have supported my research in geometric group theory, discrete groups, and dynamical systems.

2. Guggenheim Fellowship:
One of three awards in pure mathematics in 2003, this $35,000 fellowship was used to support a second semester of my sabbatical, in 2003-04, at the Institute for Advanced Study in Princeton, N.J. The grant is based entirely on merit, and does not require the grantee to pursue any particular line of investigation, though it is expected that the grantee would engage in creative work related to their field. I used the grant to study an assortment of problems in geometry and dynamics. (The Institute for Advanced Study also awarded me a research fellowship for the same purpose and roughly the same amount, but withdrew the support in deference to the Guggenheim Fellowship.)

Spherical CR Geometry and Dehn Surgery (research monograph) Annals of Mathematics Studies

Complex Hyperbolic Triangle Groups, Proceedings of the International Congress of Mathematicians

Real Hyperbolic on the Outside, Complex Hyperbolic on the Outside Inventiones Mathematicae

Ideal Triangle Groups, Dented Tori, and Numerical Analysis, Annals of Math

Symmetric Patterns of Geodesics and Automorphisms of Surface groups inventiones mathematicae

Quasi-Isometric Rigidity and Diophantine Approximation Acta Mathematica

The quasi-isometry classification of rank one lattices Publications IHES

Pappus's Theorem and the Modular Group Publications IHES

National Merit Scholarship, 1984-87

National Science Foundation Graduate Fellowship, 1987-90

Alfred P. Sloan Dissertation Fellowship, 1990-91

National Science Foundation Postdoctoral Fellowship, 1993-95

Alfred P. Sloan Research Fellowship, 1996-2000

Invited speaker, International Congress of Mathematicians, 2002

John Simon Guggenheim Fellow, 2003

Simons Fellow, 2012,2016

Fellow of the American Math Society

Fellow of the American Math Society

geometry, algebra, analysis, topology, differential geometry, combinatorics, graph
theory, infinite groups, manifolds