Geometric and Probabilistic Problems from Low Dimensional Gauge Theories

这个项目致力于解决量子物理产生的数学问题，特别是研究物质的基本组成部分之间的力。这个项目感兴趣的领域包括杨-米尔场，它调解了自然界基本粒子之间的相互作用。这些场由连接的几何概念进行数学建模。这些都是研究引力和自然界基本粒子相互作用的基础。这种场的量子行为用泛函积分的形式进行了数学描述。研究在很大程度上是通过G.“t Hooft在20世纪70年代初的工作开始的，在用数字N描述的物理场的对称尺寸很大时，这种积分的行为中发现了大量更深层次的结构和想法。该项目提出的一个方向是开发一个数学上完整的描述，描述这个大的N个“主控领域”。一个被称为自由概率理论的领域预计将在这方面发挥核心作用。该项目旨在发现新的数学结果和结构，例如从极限大N理论中产生的广义几何，其灵感来自物理理论。数学背景从微分几何到随机学和大型随机矩阵。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。该项目将涉及国际研究合作。在更广泛的智力层面上，该项目将产生来自物理和数学前沿的富有成效的想法组合。

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.