In this project Minsky will study connections between the geometric structure of hyperbolic 3-manifolds, and the combinatorial structure of surfaces. The main focus of the project is a program to solve Thurston's Ending Lamination Conjecture, which states that a hyperbolic 3-manifold is uniquely determined by its topological type and a list of invariants describing the asymptotic geometry of its ends. This conjecture is central in the field and would settle a number of outstanding questions about the structure of the deformation space of all hyperbolic structures on a given manifold. The principal tool in this approach is the Complex of Curves, a combinatorial object which describes the set of all essential homotopy classes of simple loops on a given surface. Such a complex, viewed as a metric space, is hyperbolic in the sense of Cannon and Gromov, and the invariants which describe the geometry of a 3-manifold turn out to be "points at infinity" for the complex. A study of geodesics within the complex of curves then allows us to construct a model for the geometry of the 3-manifold which depends only on its end invariants. Part of the program is joint work with Brock and Canary. The complex of curves is also instrumental in studying the Teichmuller space of hyperbolic structures on a surface, and its symmetry group, known as the Mapping Class Group. Hyperbolicity of the complex implies a type of relative hyperbolicity property for the Teichmuller space and the Mapping Class Group, and in collaboration with Brock and Masur, Minsky hopes to apply this to study the geometric properties of Teichmuller space, and to answer a number of group-theoretic and algorithmic questions about the mapping class group.
Topological spaces appear in mathematics at varying levels of abstraction. The earth, for example, can be thought of as a sphere, spinning around its axis. The set of all positions of this sphere, perhaps as parameterized by the time of day in New York, can itself be visualized as a circle. The positions of the earth in its orbit around the sun also make up a circle, and all this information together can be described by a 2-dimensional surface, called a torus, so that each point in the torus corresponds to exactly one position of the earth-sun system, and our motion in space gives rise to a trajectory winding around the torus. Once a surface is being studied, we often give it various structures: it can have a metric, which is a way of measuring distance in it; it can also be filled up in different ways by loops. The set of all possible metrics can itself be studied as a "space", where motion in this space corresponds to changing shapes of the surface. The set of all loops in the surface gives rise to another space, of a more combinatorial character. A fascinating phenomenon, not uncommon in mathematics, is the appearance of analogies between these different levels of abstraction. For example, a property of a single metric on a surface called "hyperbolicity" appears again, in another guise, as a property of the space of all metrics on the surface and the space of all loops. The interaction between these different levels yields interesting phenomena and tools which can be applied to solve a number of problems in
