Mathematical Sciences: Methods of Theoretical Physics in Topology

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This project makes use of recent ideas in theoretical physics, such as gauge theory and topological quantum field theories, to study the topology of low-dimensional manifolds. Topics to be considered include (1) Floer homology and 4-manifold invariants, (2) magnetic monopoles and polynomial invariants for 3-manifolds, (3) topology and symmetric hyperbolic equations, (4) the geometry of magnetic monopoles. There is a long tradition of mathematicians involving themselves in the problems of physics, often to the mutual enrichment of both subjects. Tools are developed to solve physical problems and then turn out to have much greater generality and become widely used in mathematics. The story of the new quantum invariants of 4-manifolds which figure in this project fits this pattern. Mathematicians interesting themselves in problems of quantum field theory were led to a certain geometric construction. The construction led in turn to an invariant that depended only on the topological character and not the full geometric character of the underlying manifold. There are now variations on the original construction and numerous resulting quantum invariants. Their origin is sufficiently different from that of other previously known invariants that they can be expected to detect things the traditional invariants cannot, but it behooves topologists to demonstrate this explicitly, as well as to sort out the different quantum invariants and their relationships to more traditional invariants.