Geometric and Probabilistic Problems from Low Dimensional Gauge Theories

低维规范理论的几何和概率问题

• 批准号: 0601141
• 负责人: Ambar Sengupta
• 金额: $ 12.57万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601141&HistoricalAwards=false
• 关键词: Geometric; Probabilistic; Problems; Low; Dimensional

This project is devoted to mathematical problems arising from quantum physics, in particular from the study of forces between the fundamental constituents of matter. The fields of interest for this project include Yang-Mills fields, which mediate interaction between fundamental particles of nature. These fields are modeled mathematically by the geometric notion of connections. These are fundamental to the study of both gravitation and the interactions of elementary particles of nature. The quantum behavior of such fields is described mathematically in terms of functional integrals. Investigations, begun in large part through the work of G. "t Hooft in the early 1970s, have found a wealth of deeper structures and ideas in the behavior of such integrals when the physical field's symmetry size, described by a number N, is large. One direction being proposed in the project is the development of a mathematically complete description of this large-N 'master field.' An area known as free probability theory is expected to play a central role here. Other directions, involving generally the same areas of mathematics, will also be pursued.The project aims to discover new mathematical results and structures, such as a generalized geometry arising from the limiting large-N theory, which are inspired by physical theories. The mathematical contexts range from differential geometry to stochastics and large random matrices. The research work on this project is expected to have a positive impact on graduate and undergraduate education through seminars and lectures to be arranged in the course of the project. The project would involve international research collaboration. In a broader intellectual plane, the project would produce a fruitful combination of ideas from the frontiers of physics and mathematics.

这个项目致力于数学 量子物理学中出现的问题，特别是研究物质基本成分之间的力。该项目感兴趣的领域包括杨米尔斯领域， 自然界基本粒子之间的相互作用这些领域的数学模型的几何概念的连接。这些都是研究引力和 自然界基本粒子的相互作用。这种场的量子行为在数学上用以下术语描述： 泛函积分 调查，开始在很大程度上通过工作的G。t Hooft等人在20世纪70年代早期，在这类积分的行为中发现了丰富的更深层次的结构和思想 当物理场的对称尺寸（用数字N来描述）很大时。该项目提出的一个方向是发展一个数学上完整的描述这个大N '主字段。“一个被称为自由概率论的领域预计将在这里发挥核心作用。本项目的目标是发现新的数学结果和结构，例如受物理学理论启发的由极限大N理论产生的广义几何。数学内容从微分几何到随机和大型随机矩阵。 该项目的研究工作预计将通过在项目过程中安排的研讨会和讲座对研究生和本科生教育产生积极影响。的 该项目将涉及国际研究合作。中 在更广阔的智力领域，该项目将产生一个富有成效的结合思想从前沿的物理学和数学。