Critical Points of Functions, Multidimensional Hypergeometric Integrals, and Quantum Integrable Systems

超几何函数是由伦哈德·欧拉在18世纪引入和研究的。该函数的现代版本出现在不同的数学和物理理论中，包括表示论、代数几何、规范理论和统计力学，并在这些不同的研究领域中从不同的角度进行考虑。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，并应用于这些不同的理论。多维(q-)超几何积分及其半经典极限、Bethe特征函数和特征向量是量子可积系统、表象理论、代数几何、规范理论和统计力学中微分方程式和差分方程的解。这些方程和解具有丰富的数学结构。多维超几何积分提供了一种将这些理论的对象和结构转化为几何对象和结构的方法，并提供了与积分相关的主函数和权函数的分析。该项目的目标是发展这一分析和几何学与上述理论的应用。该项目包括研究量子群的表示、量子可积系统的哈密顿代数、量子上同调及其相关的量子微分方程、Frobenius结构、Bethe ansatz方法、超平面排列理论和函数临界点的奇异性理论。研究计划如下：1)构造部分旗簇余切丛等变量子微分方程解的q-超几何解。2)用伴随的超几何(或q-超几何)主函数的临界集上的函数代数来确定部分标志簇的余切丛的量子上同调代数。证明了椭圆Bethe代数与相应椭圆主函数的临界集上的函数代数是一致的；发展了与椭圆Schubert演算的关系。证明了XXX Bethe代数与适当变形的Schubert簇的交点上的函数代数。3)在KZ型连接的基础上，找出KZ型连接的势和Frobenius类结构。4)建立与仿射李代数相关的主函数的临界集与与该李代数相关的经典可积族之间的关系。建立与单李代数上不可约模的张量积相关的主函数的临界点群体与张量积到不可约的分解之间的对应关系。5)利用判别安排，给出了单李代数上不可约模的BGG归结的几何实现。

The hypergeometric function was introduced and studied in the 18th century by Leonhard Euler. Modern versions of that function appear in different mathematical and physical theories, including representation theory, algebraic geometry, gauge theory, and statistical mechanics, and are considered from different points of view in these various research domains. The goal of this project is to develop a unified analysis and geometry of modern multidimensional hypergeometric functions with applications to these different theories. The work will lead to better understanding of interrelations between those parts of mathematics and physics as well as to establishing new connections among them.Multidimensional (q-)hypergeometric integrals and their semiclassical limits, Bethe eigenfunctions, and eigenvectors appear as solutions to differential and difference equations in quantum integrable systems, representation theory, algebraic geometry, gauge theory, and statistical mechanics. The equations and solutions have rich mathematical structures. The multidimensional hypergeometric integrals provide a way to transform the objects and structures of those theories to objects and structures of geometry and analysis of master functions and weight functions associated with the integrals. The goal of the project is to develop this analysis and geometry with applications to the above theories. The project involves study of representations of quantum groups, algebras of Hamiltonians of quantum integrable systems, quantum cohomology and associated quantum differential equations, Frobenius structures, the Bethe ansatz method, theory of arrangements of hyperplanes, and singularity theory of critical points of functions. The research plan is as follows: 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. Identify the elliptic Bethe algebra with the algebra of functions on the critical set of the corresponding elliptic master function; develop relations with an elliptic Schubert calculus. Identify the XXX Bethe algebra with the algebra of functions on the intersection of suitably deformed Schubert varieties. 3) Find a potential for a KZ-type connection and a Frobenius-like structure on the base of the KZ-type connection. 4) 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. Develop a correspondence between populations of critical points of master functions associated with a tensor product of irreducible modules over a simple Lie algebra and the decomposition of the tensor product into irreducibles. 5) In terms of discriminantal arrangements develop a geometric realization of the BGG resolution of an irreducible module over a simple Lie algebra.