Multidimensional Hypergeometric Integrals, Quantum Differential Equations, and Integrable Systems

数学分析在十八世纪的发展导致了除多项式函数和三角函数之外的新的特殊函数的发展。特别是，著名的超几何函数介绍和研究在十八世纪的莱昂哈德欧拉，然后由领先的数学家，他们的时间：高斯，雅可比，库默，富克斯，黎曼，施瓦茨，克莱因。该函数的现代版本出现在不同的数学和物理理论中（如表示论、代数几何、规范理论、统计力学），并从不同的角度在这些理论中被考虑。这个项目的目标是发展一个统一的分析和现代多维超几何函数的几何与应用上述理论。它将导致更好地理解数学和物理学的这些部分之间的相互关系，以及建立它们之间的新联系。该项目包括研究生的培养，该项目涉及量子群的表示、量子上同调和相关的量子微分方程、KZ型微分和差分方程、量子可积系统的Hamilton代数、Bethe代数方法和超几何函数的研究。 特别是首席研究员计划：1）构建量子微分方程和不同类型的微分和差分KZ方程的解的可积表示，并应用于三维镜像对称; 2）研究这些方程的约化及其解模素数;第三章研究相关量子可积系统中的交换哈密顿量的频谱。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估而被认为值得支持和更广泛的影响审查标准。

The development of mathematical analysis in the eighteenth century led to the development of new special functions in addition to polynomial functions and trigonometric functions. In particular, the famous hypergeometric function was introduced and studied in the eighteenth century by Leonhard Euler and then by the leading mathematicians of their time: Gauss, Jacobi, Kummer, Fuchs, Riemann, Schwarz, and Klein. Modern versions of that function appear in different mathematical and physical theories (such as representation theory, algebraic geometry, gauge theory, statistical mechanics) and are considered in these theories from different points of view. The goal of this project is to develop a unified analysis and geometry of modern multidimensional hypergeometric functions with applications to the above theories. It will lead to a better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them. The project includes the training of graduate students.This project involves research on representations of quantum groups, quantum cohomology and associated quantum differential equations, KZ type differential and difference equations, algebras of Hamiltonians of quantum integrable systems, Bethe ansatz method, and hypergeometric functions. In particular the principal investigator plans to: 1) construct integrable representations for solutions of the quantum differential equations and different types of differential and difference KZ equations with applications to the three dimensional mirror symmetry; 2) study the reduction of these equations and their solutions modulo a prime number; 3) study of the spectrum of commuting Hamiltonians in the associated quantum integrable systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.