Mathematical Sciences: Gauge Theory on Compact Surfaces

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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. ***