Mathematical Sciences: Methods of Theoretical Physics in Topology

该项目利用了理论物理学的最新思想， 例如规范理论和拓扑量子场论， 研究低维流形的拓扑。 议题 所考虑的包括（1）Floer同调和4-流形不变量， (2)磁单极子和3-流形的多项式不变量， (3)拓扑与对称双曲型方程;（4）几何 磁单极子。 有一个悠久的传统，数学家涉及 在物理问题上，他们自己，往往是相互的， 丰富了这两个主题。 开发工具来解决物理问题 问题，然后变成有更大的普遍性， 在数学中被广泛使用。 新量子的故事 不变量的4-流形图在这个项目适合这个 格局 数学家们热衷于 量子场论被引向某种几何结构。 这种构造又导致了一个不变量，它只依赖于 的拓扑性质，而不是完整的几何性质， 底层的流形 现在有了一些变化 构造和许多由此产生的量子不变量。 他们的 起源与以前的其他起源完全不同。 已知的不变量，他们可以预期检测的东西， 传统的不变量不能，但它理应拓扑学家 明确地证明这一点，以及整理出不同的 量子不变量及其与更传统的 不变量

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.