Instantons, low dimensional topology and knotted graphs

高能物理学家认为，杨-米尔斯方程模拟了夸克的行为。事实证明，杨-米尔斯方程具有非常丰富的数学结构。这些方程使得用其他方法无法获得的时空模型的研究成为可能，并为研究DNA的拓扑结构提供了工具。PI将继续研究杨-米尔斯方程及其与数学的许多不同部分的联系，研究非线性偏微分方程、低维拓扑学、代数几何、表示论和图论。研究了三个流形及其结图的Floer同调不变量的PI。特别是与彼得·克朗海默一起，PI正在研究一族Floer同调理论，这些理论是通过三个流形中的结图与指定奇点的联系建立起来的。这些理论的某些版本似乎与泡沫的Khovanov-Rozansky SL(N)-同调有关，这种关系将被探索和阐明。其他版本指向比Khovanov-Rozansky同调更一般的理论。另一个相当微妙的版本似乎与空间图的三色性问题有关，特别是可能为平面图的三色性问题提供新的见解。

High energy physicists believe that the Yang-Mills Equations model the behavior of quarks. The Yang-Mills Equations turn out to have a remarkably rich mathematical structure. These equations enable the study of models of space-time inaccessible by other means and in addition give tools for the study of the topological structure of DNA. The PI will continue his research on the Yang-Mills equations and their connections to many different parts of mathematics, the study of nonlinear partial differential equations, low dimensional topology, algebraic geometry, representation theory and graph theory. The PI with study Floer homology invariants for three manifolds and knotted graphs in them. In particular with Peter Kronheimer, the PI is studying a family of Floer homology theories built from connections with a prescribed singularity along knotted graphs in three manifolds. Certain versions of these theories appear to be related to Khovanov-Rozansky SL(N)-homology for foams and the relations will be explored and elucidated. Other versions point to theories more general than Khovanov-Rozansky homology. Yet another rather delicate version appears to have bearing on questions of tri-colorability of spacial graphs, and in particular appears likely to provide novel insight to the question of tri-colorability of planar graphs.