Quantum Cohomology, Representation Theory, and Feynman Amplitudes

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Principal Investigator: Prakash BelkaleProposal Number: 0300356Institution: University of North Carolina at Chapel HillAbstract: Quantum Cohomology, Representation theory and Feynman AmplitudesThe principal investigator wants to pursue research in two areas: Representations of the fundamental group and Quantum Cohomology, and the study of the structure of Feynman amplitudes (the second area is in collaboration with Patrick Brosnan of UCLA). In the first field, the PI wants to generalise his recent geometric proof of Horn and Saturation conjectures to the Quantum analogues of these conjectures. This project is inspired by the the problem of unitary representations of the fundamental group of p1 of projective n-space with n points removed with prescribed local monodromies. The principal investigator plans to investigate the existence of Horn type recursion for other groups and study analogues of the Saturation conjecture. A final goal of this work is to determine an optimal set of inequalities for the problem of existence of unitary representations with prescribed monodromies. In the second field (in collaboration with Brosnan), the principal investigator will continue the study of relations between Feynman amplitudes and algebraic geometry. The first step of this work is to understand in general Algebro-geometric terms the integral computations of the physicists.The study of relations between Representation theory (`symmetries') and Algebraic Geometry is a very important area of research. Part of the motivation for this work comes from eigenvalue problems, which are important in numerical computing and in wave mechanics. The work on the geometry of Feynman amplitudes is of interest both in mathematics and physics. The aim is a better mathematical understanding of Feynman amplitudes, which are fundamental to the quantum theory.