Instantons, low dimensional topology and knotted graphs

瞬子、低维拓扑和打结图

• 批准号: 1406348
• 负责人: Tomasz Mrowka
• 金额: $ 47.63万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406348&HistoricalAwards=false
• 关键词: Instantons; low; dimensional; topology; knotted

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.

高能物理学家认为杨-米尔斯方程模拟了夸克的行为。杨-米尔斯方程具有非常丰富的数学结构。这些方程使研究模型的时空无法通过其他手段，此外还提供了工具的研究拓扑结构的DNA。PI将继续他的研究杨米尔斯方程及其连接到数学的许多不同部分，非线性偏微分方程的研究，低维拓扑，代数几何，表示论和图论。 PI研究了三个流形和其中的纽结图的Floer同调不变量。特别是与彼得Kronheimer，PI正在研究一个家庭的弗洛尔同源理论建立的连接与一个规定的奇点沿着打结的图形在三个流形。 这些理论的某些版本似乎与泡沫的Khovanov-Rozansky SL（N）-同源性有关，并将探讨和阐明这种关系。其他版本指向比Khovanov-Rozansky同源性更普遍的理论。 然而，另一个相当微妙的版本似乎对空间图的三色性问题有影响，特别是似乎有可能提供新的见解，平面图的三色性问题。