3-Manifold Topology, Character Varieties and Dehn Surgery
Xingru Zhang Principal Investigator
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Proposal: DMS-9971561. Principal Investigator: Xingru Zhang. Abstract: The main theme of this project is to understand the topology of hyperbolic knots or links in 3-manifolds and its variation under Dehn surgery through studying the SL(2,C)-character varieties associated to the knot or link complements. Applications to exceptional Dehn surgery,especially to the property P conjecture and the cabling conjecture will be explored. It is well known that all closed orientable 3-manifolds are interrelated topologically through Dehn surgery on knots and links. This relation is nicely and deeply encoded algebraically in the SL(2,C)-character varieties of the involved knot or link complements. Studying the connections between the topology of 3-manifolds and their character varieties has been very successful in understanding 3-manifolds and their change under Dehn surgery. This project is a continuation of this line of investigation, with an eye to possible solutions of some outstanding conjectures. 3-manifold topology is a branch of mathematics in which we study and classify certain global structures of "universes" (called 3-manifolds by mathematicians). It is natural and compelling for human beings to understand 3-manifolds because we believe we live in one of them. Besides being interesting and beautiful in its own right, 3-manifold topology has profound connections and interactions with other branches of mathematics, and also with physics, chemistry, biology and computer science. It is one of most active and fertile research fields in modern mathematics. The current project aims at solving several fundamental problems in 3-manifold topology.