Mathematical Sciences: The Topology of Three-Manifolds

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Frohman intends to pursue the study of three-dimensional manifolds, especially topics associated with Heegaard splittings and minimal surfaces and the representation theory of knot groups and three-manifold groups. For instance, he is planning to study the finiteness of Heegaard splittings of bounded genus of three- manifolds up to homeomorphism. Another problem he is pursuing is the construction of functors that are similar to coupled topological quantum field theories using the intersection homology of moduli spaces of semistable holomorphic bundles with parabolic structure over a nonsingular algebraic curve. Three-dimensional manifolds are very natural objects for topologists to study, since (ignoring time) we live in a physical world of three dimensions. Also dimension three is the first dimension where many types of rich complexity arise. In fact, it is a notorious and paradoxical fact that some of the difficulties presented by three-dimensional topology disappear again in higher dimensions. For example, the celebrated Poincare conjecture about the characterization of the three-dimensional sphere remains as challenging as when it was first posed about the turn of the century, while the analogous propositions in higher dimensions have all now been settled. Frohman is exploring some of these mysteries and sharpening the available tools of the trade.