Multidimensional Hypergeometric Integrals, Quantum Differential Equations, and Integrable Systems

18世纪数学分析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，著名的超几何函数是由伦哈德·欧拉在18世纪引入和研究的，然后是他们那个时代的著名数学家：高斯、雅各比、库默、富克斯、黎曼、施瓦兹和克莱因。该函数的现代版本出现在不同的数学和物理理论(如表示理论、代数几何、规范理论、统计力学)中，并从不同的角度在这些理论中进行考虑。这个项目的目标是发展现代多维超几何函数的统一分析和几何，并应用于上述理论。它将导致更好地理解数学和物理的这些部分之间的相互关系，并在它们之间建立新的联系。这个项目包括对研究生的培训，这个项目涉及到量子群的表示，量子上同调及其相关的量子微分方程，KZ型微分方程组和差分方程，量子可积系统的哈密顿代数，Bethe ansatz方法，以及超几何函数。具体地说，主要研究人员计划：1)构造量子微分方程解的可积表示，并应用于三维镜像对称性；2)研究这些方程及其解的模为素数的约化；3)研究相关量子可积系统中交换哈密顿量的谱。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.