Critical Points of Master Functions, Hypergeometric Integrals of Arrangements, and Quantum Integrable Systems

超几何函数是世纪由欧拉提出并研究的。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），并从不同的角度考虑这些理论。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。该项目涉及量子群的表示、量子可积系统的哈密顿代数、量子上同调和相关的量子微分方程、Frobenius结构、Bethe anomaly方法、超平面排列理论、函数临界点的奇点理论特别是主要研究者计划：1）构造部分旗簇的余切丛的等变量子微分方程的q-超几何解; 2）识别部分旗簇的余切丛的量子上同调代数与相关超几何（或q-超几何）主函数的临界集上的函数代数; 3）。利用qKZB方程的超几何解定义了椭圆动力量子群对部分旗簇的椭圆等变上同调的作用，4）在KZ型联络的基础上找到了一个KZ型联络的势和一个Frobenius型结构;第五章）建立与仿射李代数相关的主函数的临界集和与该李代数相关的经典可积族之间的关系代数; 6）利用分圆判别安排构造分圆KZ方程的超几何解：7）利用某些类型李代数之间的折叠关系，将一类Bethe代数几何化。

The hypergeometric function was introduced and studied in 18th century by Leonhard Euler. Modern versions of that function appear in different mathematical and physical theories (like 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 better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them.This project involves research on representations of quantum groups, algebras of Hamiltonians of quantum integrable systems, quantum cohomology and associated quantum differential equations, Frobenius structures, Bethe ansatz method, theory of arrangements of hyperplanes, singularity theory of critical points of functions. In particular the principal investigator plans to: 1) construct q-hypergeometric solutions of the equivariant quantum differential equation for the cotangent bundle of a partial flag variety; 2) identify the quantum cohomology algebra of the cotangent bundle of a partial flag variety with the algebra of functions on the critical set of the associated hypergeometric (or q-hypergeometric) master function; 3). use hypergeometric solutions of qKZB equations to define an action of elliptic dynamical quantum groups on elliptic equivariant cohomology of partial flag varieties; 4) find a potential for a KZ-type connection and a Frobenius-like structure on the base of the KZ-type connections; 5) develop a relation between the critical set of master functions associated with an affine Lie algebra and classical integrable hierarchies associated with that Lie algebra; 6) construct hypergeometric solutions of cyclotomic KZ equations in terms of cyclotomic discriminantal arrangements; 7) use the folding relation between the Lie algebras of certain types to geometrize a type of Bethe algebra.