Mathematical Sciences: Gauge Theory on Compact Surfaces

[400961] Sengupta该项目涉及对紧致曲面上的规范理论（或杨米尔斯理论）引起的几个数学问题的研究。第一个项目是在无限维非线性空间（代表规范场物理概念的某些几何对象）上构造某个测度；下一个目标是解决与这项措施有关的一些具体问题。其中一些问题涉及到“平面连接的模空间”，从其他角度来看，这个空间的结构在数学中一直很有趣，并且涉及到一些深层次的问题。规范理论被用来描述物质基本成分之间的相互作用。虽然这些理论起源于物理学，但它们涉及数学中许多优雅而深刻的几何思想。当与量子场论相结合以获得对自然的更现实的描述时，规范理论产生了更复杂的数学问题。即使在没有直接物理意义的情况下，这些问题也具有内在的数学价值。本建议处理这种性质的数学问题。这些问题涉及到各种数学学科的思想，包括拓扑、几何和概率论。＊＊＊

9400961 Sengupta The project involves an investigation of several mathematical problems arising from gauge theory (or Yang Mills theory) on compact surfaces. The first project is to construct a certain measure on an infinite dimensional non-linear space (of certain geometric objects which represent the physical concept of gauge fields); the next objective is to solve a number of concrete problems related to this measure. Some of these questions involve the "moduli space of flat connections", a space whose structure has been of interest in mathematics from other points of view and has involved deep questions. Gauge theories are used to describe the interactions of the fundamental constituents of matter. While these theories arise in physics, they involve many elegant and deep geometric ideas in mathematics. When combined with quantum field theory to obtain a more realistic description of nature, gauge theories give rise to more complex mathematical questions. Such questions are of intrinsic mathematical value even in situations where there is no immediate physical significance of the question. The present proposal addresses mathematical problems of this nature. The problems involve ideas from a variety of mathematical disciplines, including topology, geometry and probability theory. ***