New tools for gauge theory in dimensions 3 and 4

高能物理学家认为杨-米尔斯方程模拟了夸克的行为--夸克是物质的基本组成部分。 杨-米尔斯方程有着非常丰富的数学结构，除了直接应用于物理学之外，它还使得时空模型的研究无法通过其他方法进行。它们甚至为研究DNA的拓扑结构提供了工具。PI将继续他对杨米尔斯方程的研究，以及相关的想法，重点是它与数学的许多不同部分的联系，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。该奖项为从事相关研究的学生提供支持。 PI将研究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。这些理论在低维拓扑问题中有许多应用。PI计划将Instanton Floer同源性的一个关键缺失功能放在适当的位置-为所有三个流形提供完整的“等变”版本。这将使新的计算和产生新的应用。PI将开发工具，用于对代数几何和数论中与Land和Caporaso-Harris-Mazur有关的一些基本问题进行拓扑攻击。然而，另一个相当微妙的版本似乎与空间图的三色性问题有关，特别是似乎可能为平面图的三色性问题提供新的见解。该奖项反映了NSF的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

High energy physicists believe that the Yang-Mills Equations model the behavior of quarks - the fundamental constituents of matter. The Yang-Mills Equations turn out to have a remarkably rich mathematical structure which, outside of direct applications to physics, enable the study of models of space-time inaccessible by other means. They even give tools for the study of the topological structure of DNA. The PI will continue his research on the Yang-Mills equations, and related ideas focusing on its 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 award provides support for students engaged in related research. The PI will 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. These theories have many applications to questions in low dimensional topology. The PI plans to put a key missing feature for Instanton Floer homology into place-- having a full “equivariant” version available for all three manifolds. This should enable new computations and give rise to new application. The PI will develop tools for a topological attack on some basic questions in Algebraic Geometry and Number theory related to conjectures of Land and Caporaso-Harris-Mazur. Yet another rather delicate version appears to have bearing on questions of tri-colorability of spacial graphs in particular appears likely to provide novel insight to the question of tri-colorability of planar graphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.