Surfaces in low-dimensional topology

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The investigator proposes to study the topology of 3-dimensional manifolds,using 2-dimensional laminations and foliations as the basic tools. The maingoal is to show how these objects can be used to attack several long-standingconjectures in low-dimensional topology. In particular, he proposes to useconstructions of essential laminations, due to the investigator and others,to classify the exceptional Dehn surgeries on alternating knots. Heplans to use laminations to show that some large classes of 3-manifoldssatisfy the Virtual Haken Conjecture. He will also continue work aimed at providing a better understanding of the structure of sutured manifolddecompositions, a fundamental technique in building and using taut foliations,by developing examples of hyperbolic knots with large depth.We live in and move through a 3-dimensional universe, which we perceiveas extending infinitely in all directions. But recent observations hint at thepossibility that space may instead wrap around itself, so that we live in a finite universe. It is this very idea of a space wrapping around itself that lies at theheart of geometric topology, whose goal is to understand the global structure of objects which locally look like ordinary Euclidean space. The investigator plans to study several questions aimed at uncovering patterns in the global structure of 3-dimensional spaces, by using 2-dimensional surfaces to study the spaces in which they can be found. The main idea is to use surfaces as away of cutting the space into pieces; by studying the resulting pieces, we can gain insight into the large-scale structure of the 3-dimensional space.