Quantum Cohomology, Representation Theory, and Feynman Amplitudes

主要研究人员：Prakash BelkaleProposal Number：0300356机构：北卡罗来纳大学教堂山分校摘要：量子上同调，表示理论和费曼放大主要研究两个领域：基本群和量子上同调的表示，以及费曼幅度结构的研究(第二个领域是与加州大学洛杉矶分校的帕特里克·布罗斯南合作)。在第一个领域，PI想要将他最近对Horn猜想和饱和猜想的几何证明推广到这些猜想的量子类似。这个项目的灵感来自于n维射影空间中n点被指定的局部单行去掉的射影n-空间p1的基本群的么正表示问题。主要研究人员计划研究其他群的Horn型递推的存在性，并研究饱和猜想的类似情况。这项工作的最终目标是为具有指定单行的酉表示的存在性问题确定一组最优的不等式。在第二个领域(与布鲁斯南合作)，首席研究员将继续研究费曼振幅和代数几何之间的关系。这项工作的第一步是用一般的代数几何术语理解物理学家的积分计算。研究表示论(对称性)和代数几何之间的关系是一个非常重要的研究领域。这项工作的部分动机来自于本征值问题，这在数值计算和波浪力学中是重要的。费曼振幅的几何研究在数学和物理学中都很有趣。其目的是更好地从数学上理解费曼振幅，这是量子理论的基础。

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.