Research Interests

My research concerns the topology and geometry of two and three-dimensional manifolds. An n-dimensional manifold is an object that locally looks like the standard Euclidean n-space. For example the sphere and the torus (the surface of the donut) are examples of 2-dimensional manifolds. When a manifold has extra structure, such as a taut foliation or a hyperbolic metric (i.e. a metric of constant negative curvature) a beautiful interplay between the geometry and topology often ensues. A 3-manifold has a taut foliation if it can be decomposed into a coherent union of infinitely thin sheets (somewhat like an onion) such that all the sheets can in an appropriate sense be pulled tight. I proved that many 3-dimensional manifolds have taut foliations and used that to prove the Property R conjecture in knot theory. I.e. there is an essentially unique way to go from the 3-sphere to the product of the sphere and the circle by drilling out one solid tube from each. Others have used my work on taut foliations to establish fundamental properties in the theory of contact structures, Heegaard Floer homology and instanton homology. I proved the Smale conjecture for hyperbolic 3-manifolds which says that the space of diffeomorphisms is homotopy equivalent to the space of isometries. Danny Calegari and I (and independently Ian Agol) proved Marden's tameness conjecture which asserts that for hyperbolic 3-­manifolds, the part of the manifold near "infinity" is topologically as standard as possible. Tameness completed the proof of the Ahlfors measure conjecture and is crucial for the resolution of many other important conjectures in hyperbolic geometry. Rob Meyerhoff, Peter Milley and I found the smallest hyperbolic 3-manifold. Roughly speaking we found the most efficient way of symmetrically packing hyperbolic 3-space by balls.

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Section 11: Mathematics