New tools for gauge theory in dimensions 3 and 4

3 维和 4 维规范理论的新工具

• 批准号: 2105512
• 负责人: Tomasz Mrowka
• 金额: $ 49.35万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105512&HistoricalAwards=false
• 关键词: New; tools; gauge; theory; dimensions

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.

高能量物理学家认为，阳米尔方程对夸克的行为进行了建模 - 物质的基本结构。 Yang-Mills方程证明具有非常丰富的数学结构，除了直接应用物理学外，还可以通过其他方式研究时空无法访问的模型。他们甚至提供了研究DNA拓扑结构的工具。 PI将继续对Yang-Mills方程的研究，以及关注其与数学的许多不同部分的联系，非线性偏微分方程，低维拓扑，代数几何，表示理论和图形论的研究。该奖项为从事相关研究的学生提供了支持。 PI将研究三个流形和打结图的浮动同源性不变性。尤其是与彼得·克朗海默（Peter Kronheimer）一起，PI正在研究一种浮点同源性理论，该家族是由连接构建的，并在三个歧管中沿着打结的图形沿规定的奇异性建立。这些理论对低维拓扑的问题有许多应用。 PI计划为Instanton Floer同源性制作关键的缺少功能 - 所有三个歧管都有完整的“ eproivariant”版本。这应该启用新的计算并引起新的应用。 PI将开发工具，以对代数几何学的一些基本问题以及与土地和Caporaso-Harris-Mazur的猜想有关的数字理论进行拓扑攻击。另一个相当微妙的版本似乎与空间图的三色性问题有关，似乎有可能为平面图的三色性问题提供新颖的见解。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛影响的评估审查审查标准来通过评估来获得的支持。