Seiberg-Witten invariants of three-manifolds

项目负责人：Liviu I. nicolaesu本项目涉及三维Seiberg-Witten单极子理论的拓扑和几何方面，并解决了两个一般性的指导问题。第一个问题是关于单极子数的拓扑意义。Mengand Taubes证明了在具有非平凡同调的三流形上，这个计数是由Reidemeister-Turaev扭转给出的。在有理同调球的情况下，seifert流形的数值实验表明，适当改变单极子数以非常优雅的方式决定了Casson-Walker不变量和Reidemeister-Turaev扭转。该项目的第一部分致力于证明这一点。第二个问题是关于三维单极子的几何意义。复杂表面奇点的链接被赋予了自然的CR几何形状，并且很自然地要问单极子捕获了多少。更准确地说，我们解决了以下基本问题：当奇点的连接坍缩到奇点分辨率的例外除数上时，domonpole“看起来像”什么？三维世界一直是代数拓扑学许多重要发展的灵感来源。它在我们的物理直觉范围内，所以一些可视化是可能的（想象一个立方体），但也有很多惊喜的空间（想象把立方体的相反面成对地粘在一起）。一种现代观点认为，如果在有利的光线下呈现，三维世界将揭示其奥秘。数学家们用一种更冷静的方式表达了这一点。如果一个人能在三流形上找到一个好的几何形状，那么他就能解开它的一些拓扑特征。单极子是物理学家引入数学世界的几何物体。人们可以把它们看作是三维空间产生的“信号”。这些“信号”取决于我们如何看待流形。自从它们出现在数学舞台上，它们就为我们提供了关于低维世界结构的惊人见解。这个项目是关于以一种有意义的方式计算这些“信号”，然后在有利的光线下观察每一个信号的意义。

Proposal DMS-0071820 Principal Investigator: Liviu I. NicolaescuThis project involves topological and geometric aspects of thetheory of three-dimensional Seiberg-Witten monopoles and itaddresses two general guiding questions. The first question isabout the topological significance of the monopole count. Mengand Taubes have shown that on a three-manifold with nontrivialhomology this count is given by Reidemeister-Turaev torsion. Inthe case of rational homology spheres numerical experiments withSeifert manifolds suggest that a suitably altered monopole count determine in a very elegant fashion boththe Casson-Walker invariant and the Reidemeister-Turaev torsion.The first part of the project is devoted to proving this. Thesecond question is about the geometric meaning ofthree-dimensional monopoles. The link of a complex surfacesingularity is endowed with a natural CR geometry and it isnatural to ask how much of it does a monopole capture. Moreprecisely we address the following fundamental question: how domonopoles "look like" as the link of the singularity collapsesonto the exceptional divisor of a resolution of the singularity?The three-dimensional world has been a source of inspirationfor many important developments in algebraic topology. It iswithin the reach of our physical intuition so some visualizationis possible (think of a cube) yet there is plenty of room forsurprises (think of gluing in pairs the opposite faces of thecube). One modern point of view is that, if presented in afavorable light, the three-dimensional world will reveal itsmysteries. The mathematicians phrase this in a more sober way.If one can find a nice geometry on a three-manifold then one canunlock some of its topological features. The monopoles aregeometric objects introduced to the mathematical world byphysicists. One could think of them as "signals" produced by athree-dimensional space. These ``signals'' depend on how we lookat the manifold. Since they appeared on the mathematical scenethey have provided us with surprising insights about the structureof low-dimensional worlds. This project is about counting these"signals" in a meaningful way and then understanding thesignificance of each one of them individually when observed undera favorable light.