Floer Homology and Homology Cobordisms

这个项目是关于有向同调3球Y的Yang-Mills和Seiberg-Witten Floer同调群的。在他早期关于理性Yang-Mills Floer理论的工作中，主要研究者已经证明了Y上不可约的平坦连接和平凡连接之间的相互作用可以用一个整数来描述。该整数是Y的同调协类的不变量，是连通和下的可加性。当Y是一个具有非标准负定交点形式的光滑4流形时，不变量为正。在项目的第一部分中，主要研究者将研究相应的有理Seiberg-Witten Floer理论中的不变量，该不变量是为具有自旋-c结构的定向有理同调3球定义的。第二部分将着重于这两个不变量之间的关系，在整同调球的情况下，使用PU(2)单极子。项目的第三部分的目标将是理解由杨-米尔斯花理论产生的模2系数的同调协不变量。这些不变量在Seiberg-Witten理论中似乎没有任何类似的东西。一般来说，首席研究员将使用理论物理中的某些非线性偏微分方程研究3维和4维空间。

Proposal: DMS-9971731PI: Kim FroyshovThis project is concerned with Yang-Mills and Seiberg-Witten Floer homology groups of oriented homology 3-spheres Y. In his earlier work on rational Yang-Mills Floer theory the principal investigator has shown that interaction between irreducible flat connections and the trivial connection over Y can be described by a single integer. This integer is an invariant of the homology cobordism class of Y, and is additive under connected sums. The invariant is positive if Y bounds a smooth 4-manifold with non-standard negative definite intersection form. In the first part of the project the principal investigator will study the corresponding invariant in rational Seiberg-Witten Floer theory, which is defined for oriented rational homology 3-spheres with a spin-c structure. The second part will focus on the relationship between these two invariants in the case of integral homology spheres, using PU(2) monopoles. The object of the third part of the project will be to understand the homology cobordism invariants arising from Yang-Mills Floer theory with mod 2 coefficients. These invariants do not seem to have any analogues in Seiberg-Witten theory.Broadly speaking, the principal investigator will study spaces of dimension3 and 4 using certain non-linear partial differential equations from theoretical physics.