Multidimensional Hypergeometric Integrals, Quantum Differential Equations, and Integrable Systems

多维超几何积分、量子微分方程和可积系统

基本信息

批准号：
1954266
负责人：
Alexander Varchenko
金额：
$ 33.01万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954266&HistoricalAwards=false
关键词：
Multidimensional Hypergeometric Integrals Quantum Differential

项目摘要

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.

十八世纪数学分析的发展导致除了多项式函数和三角函数之外新的特殊函数的发展。特别是，著名的超几何函数在 18 世纪由莱昂哈德·欧拉 (Leonhard Euler) 引入并研究，随后由当时的顶尖数学家高斯、雅可比、库默、福克斯、黎曼、施瓦茨和克莱因介绍和研究。该函数的现代版本出现在不同的数学和物理理论（例如表示论、代数几何、规范理论、统计力学）中，并且在这些理论中从不同的角度进行考虑。该项目的目标是开发现代多维超几何函数的统一分析和几何，并将其应用于上述理论。它将导致更好地理解数学和物理学这些部分之间的相互关系，并在它们之间建立新的联系。该项目包括研究生培养。该项目涉及量子群表示、量子上同调及相关量子微分方程、KZ型微分和差分方程、量子可积系统哈密顿量代数、Bethe ansatz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，并将其应用于三维镜面对称； 2）研究这些方程的约简及其以素数为模的解； 3) 研究相关量子可积系统中的通勤哈密顿量谱。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和更广泛的影响审查标准进行评估，被认为值得支持。