Geometric and Probabilistic Problems from Low Dimensional Gauge Theories

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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.