Mathematical Sciences: String Equations in Mathematical Physics and Integrable Systems

本研究涉及数学物理中的弦方程和可积系统。特别地，研究涉及与量子场论和弦理论相关的偏微分方程的解，这些方程的解可以表示为特殊类型的费曼型积分。该项目还将涉及微分算子和边界的等电流形。特别地，微分算子的行为是什么以及如何在它们的爆炸轨迹附近正则化微分算子。这项研究是在几何分析和数学物理的一般领域，并说明了数学和物理之间令人兴奋的相互作用，特别是在量子场论和弦理论中，有一个涉及可积系统，模块空间，代数和矩阵模型的思想圈。

This research concerns string equations in mathematical physics and integrable systems. In particular, the research concerns the solutions of partial differential equations related to quantum field theory and string theory that may have solutions representable as special types of Feynman type integrals. The project will also concern isopectral manifolds of differential operators and boundaries. In particular, what is the behavior of the differential operators and how does one regularize the differential operators near their blow-up locus. This research is in the general area of geometric analysis and mathematical physics and illustrates the exciting interaction between mathematics and physics, especially in quantum field theory and string theory where there is a circle of ideas involving integrable systems, module space, algebras, and matrix models.