Mathematical Sciences: Atiyah's Conjectures on Floer Homology

数学科学：阿蒂亚关于弗洛尔同调的猜想

• 批准号: 9626166
• 负责人: Weiping Li
• 金额: $ 4.58万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626166&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Atiyah; Conjectures; Floer

9626166 Li The problems addressed in this project are in the area of Topology. The main theme is to study conjectures of Sir Michael Atiyah on (homology) three-spheres, using invariants obtained from mathematical physics, especially conformal field theory and string theory, and symplectic topology. The investigator and R. Lee have studied the first conjecture of Sir Michael Atiyah on identifying these invariants of three-spheres, and also extended the invariants to slightly larger classes of three-spheres. The major part of this project is, not only to identify these invariants (the first Atiyah conjecture), but also to identify the way to relate these invariants to each other (the second Atiyah conjecture). Such an identification suggests a "hidden duality" between conformal field theory and four-dimensional Yang-Mills theory. The other part of this project is to study intrinsic properties for the invariants of larger classes of three-spheres. It is a fundamental aim to investigate the change of the new invariants under certain topological operations. The project will lead to a study of relations between Floer theory, which is an invariant from mathematical physics, and other constructions of knot theory, as well as the generalization of the invariants to general three-manifolds. A three-dimensional manifold is a space where a nearsighted person sees a standard three-dimensional space everywhere. (Homology) three-spheres are those three-manifolds that one cannot tell from the standard three-sphere by using the usual topological tools. That such exist indicates the complexity of the world we live in, even ignoring the time dimension, thus making three-spheres of cosmological and physical interest. The topological invariants in the project are intended to distinguish manifolds by systematically studying their properties with respect to several quantum field theories. It is therefore valuable to investigate these new invariants and their structure s for three-spheres and general three-manifolds. This project addresses some of the most fundamental problems in this subject. ***

9626166本项目中解决的问题在拓扑领域。 主要主题是研究迈克尔·阿蒂亚爵士（Sir Michael Atiyah）对（同源性）三个角度的猜想，使用从数学物理学中获得的不变性，尤其是共形野外理论和弦理论以及符号拓扑。 研究者和R. Lee研究了迈克尔·阿蒂亚爵士的第一个猜想，以识别这些三个球的这些不变性，还将不变性扩展到了稍大的三个球体类别。 该项目的主要部分不仅是识别这些不变的人（第一个atiyah猜想），而且还要确定将这些不变性相互关联的方式（第二个Atiyah猜想）。 这样的识别表明，共形场理论和四维阳米尔斯理论之间存在“隐藏的二元性”。 该项目的另一部分是研究较大类三个球体的不变性的内在特性。 在某些拓扑操作下，调查新不变的变化是一个基本目的。 该项目将导致对浮子理论之间的关系研究，这是数学物理学的不变性，以及结理论的其他结构，以及对一般三个manifolds的不变性的概括。 三维流形是一个近视人看到到处都有标准的三维空间的空间。 （同源性）三个角度是通过使用通常的拓扑工具从标准的三个球体中无法从标准的三个球体中分辨出来的三个小球。 存在这种情况表明我们生活的世界的复杂性，甚至忽略了时间维度，从而使宇宙学和身体兴趣的三个角度。 该项目中的拓扑不变性旨在通过系统地研究其性质相对于多种量子场理论来区分流风。 因此，研究这些新的不变式及其结构S对于三个球形和一般三个manifolds是有价值的。 该项目解决了该主题中一些最根本的问题。 ***