Seiberg-Witten invariants of three-manifolds

三流形的 Seiberg-Witten 不变量

• 批准号: 0071820
• 负责人: Liviu Nicolaescu
• 金额: $ 4万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071820&HistoricalAwards=false
• 关键词: Seiberg; Witten; invariants; three; manifolds

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.

提案DMS-0071820主要研究者：Liviu I。Nicolaescu这个项目涉及三维Seiberg-Witten单极子理论的拓扑和几何方面，它解决了两个一般性的指导问题。第一个问题是关于计数的拓扑意义。孟和陶贝斯已经证明， 在具有非平凡同调的三流形上，这个计数由Reidemeister-Turaev挠率给出。在有理同调球的情况下，数值实验与塞弗特流形表明，适当改变的计数确定在一个非常优雅的时尚casson-walker不变量和Reidemeister-Turaev torsion。该项目的第一部分是致力于证明这一点。第二个问题是关于三维单极子的几何意义。一个复杂曲面的链接被赋予了一个自然的CR几何，很自然地要问它有多少是一个CR捕获的。更确切地说，我们解决以下基本问题：如何domonopoles“看起来像”作为链接的奇点abrasesonto异常除数的决议的奇点？三维世界 has been a source来源of inspiration灵感for many许多important重要developments发展in algebraic代数topology拓扑. 它在我们的物理直觉的范围内 因此，一些可视化是可能的（想想一个立方体），但也有足够的空间给惊喜（想想把立方体的相对面成对粘起来）。一种现代的观点认为，如果以一种令人愉快的方式呈现，三维世界将揭示它的神秘。数学家 用一种更清醒的方式来表达这一点，如果一个人能在一个三流形上找到一个好的几何，那么他就能解开它的一些拓扑特征。磁单极子是物理学家引入数学世界的几何对象。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以来，它们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来计数这些“信号”，然后在有利的光线下观察时，理解每一个信号的意义。