Geometric and Probabilistic Problems from Low Dimensional Gauge Theories

该项目致力于研究量子物理学中产生的数学问题，特别是研究物质基本成分之间的力。这个项目感兴趣的领域包括杨-米尔斯场，它调解自然基本粒子之间的相互作用。这些领域通过连接的几何概念在数学上建模。这些是研究引力和自然界基本粒子相互作用的基础。这些场的量子行为是用泛函积分在数学上描述的。调查在很大程度上是通过G。“t Hooft在20世纪70年代早期，在这种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对这个大n“主场”的数学完整描述。一个被称为自由概率论的领域有望在这里发挥核心作用。其他方向，一般涉及相同的数学领域，也将继续进行。该项目旨在发现新的数学结果和结构，例如由极限大n理论产生的广义几何，这些结果和结构受到物理理论的启发。数学背景范围从微分几何到随机和大随机矩阵。本项目的研究工作将通过在项目过程中安排的研讨会和讲座，对研究生和本科生的教育产生积极影响。该项目将涉及国际研究合作。在更广泛的知识层面上，该项目将产生来自物理学和数学前沿思想的富有成效的结合。

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.