Mathematical Sciences: Atiyah's Conjectures on Floer Homology

小行星9626166 在这个项目中解决的问题是在该地区的拓扑。 主要的主题是研究迈克尔·阿蒂亚爵士关于（同调）三球的结构，使用从数学物理，特别是共形场论和弦理论，以及辛拓扑学中获得的不变量。 研究者和R. Lee研究了Michael Atiyah爵士关于确定三球面不变量的第一个猜想，并将不变量扩展到稍大的三球面类。 这个项目的主要部分是，不仅要确定这些不变量（第一个Atiyah猜想），而且要确定将这些不变量相互关联的方法（第二个Atiyah猜想）。 这样的鉴别暗示了共形场论和四维杨-米尔斯理论之间的“隐藏对偶性”。 这个项目的另一部分是研究更大类的三球不变量的内在性质。 研究新的不变量在特定拓扑操作下的变化是一个基本的目的。 该项目将导致弗洛尔理论，这是一个不变量的数学物理之间的关系的研究，和其他结构的纽结理论，以及一般的三流形的不变量的推广。 一个三维流形是一个空间，一个近视的人看到一个标准的三维空间无处不在。 （同调）三球是那些三流形，人们不能从标准的三球使用通常的拓扑工具。 这种存在表明了我们所生活的世界的复杂性，即使忽略时间维度，从而使宇宙学和物理学的兴趣三个领域。 该项目中的拓扑不变量旨在通过系统地研究它们相对于几种量子场论的性质来区分流形。 因此，研究三球面和一般三流形的这些新的不变量及其结构是有价值的。 这个项目解决了这个问题中的一些最基本的问题。 ***

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. ***