Transcendental fiber functors, shift of argument algebras and Riemann-Hilbert correspondence for q-difference equations

量子群是自然界最基本对称性的变形。它们是在20世纪80年代在研究描述薄冰层的一维和二维统计力学模型时发现的。令人惊讶的是，量子群最近被证明是作为四维规范理论的对称性出现的，四维规范理论描述了夸克等基本粒子的相互作用。微分方程是科学中的另一个基本范式，描述了物理、化学、生物和经济系统的演化。它们的一个引人注目的方面是，它们可以表现出斯托克斯现象：它们的解决方案并不完全被递归捕获，并且通常用于解决它们的可编程方法。缺失的信息或斯托克斯数据可以被认为是微分方程的隐藏对称性，因为它们涉及具有相同形式展开的不同解。这个项目源于最近的发现，量子群自然产生于与经典对称性相关的微分方程的斯托克斯数据。主要目标是进一步探索经典和量子对称性之间的桥梁。特别感兴趣的是差分方程的扩展，差分方程是微分方程的自然离散化，其斯托克斯数据除了一个变量的情况外还没有得到很好的理解。另一个重要的方向将研究可积系统，或运动常数，对应于这些微分和差分方程。该项目将为研究生提供研究培训的机会。更详细地说，该项目源于量子群的超越构造，这些量子群来自于由于PI而产生的相应李代数的动力学Knizhnik-Zamolodchikov方程的Stokes数据。第一部分将构造扩展到变形参数的数值，特别是单位根和差分设置。第二部分将建立一个黎曼-希尔伯特对应的q-差分方程在几个变量，通过定义一个适当的概念，经常奇性和捕捉这些椭圆monodromy数据，类似于一个变量的情况下处理伯克霍夫。第三部分是关于由Casimir联络产生的可积系统，以及它们在相应的李代数片中的参数化。该项目的成果将在斯托克斯现象、量子可积系统和几何表示理论的研究中得到应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Quantum groups are deformations of the most basic symmetries of Nature. They were discovered during the 1980s in the study of one- and two-dimensional statistical mechanical models describing thin layers of ice. Amazingly, quantum groups have recently been shown to arise as the symmetries of 4-dimensional gauge theories, which describe the interaction of elementary particles such as quarks. Differential equations are another basic paradigm in science, and describe the evolution of physical, chemical, biological and economic systems. One of their striking aspects is that they can exhibit Stokes phenomena: their solutions are not entirely captured by the recursive, and often programmable methods used to solve them. The missing information, or Stokes data, can be considered as a hidden symmetry of the differential equation, as they relate different solutions possessing the same formal expansions. This project stems from the recent discovery that quantum groups naturally arise from the Stokes data of differential equations associated to classical symmetries. The main goals are to further explore this bridge between classical and quantum symmetries. Of particular interest is the extension to difference equations, which are natural discretisations of differential equations, and whose Stokes data are not well-understood beyond the one-variable case. Another important direction will the study of the integrable systems, or constants of motion, corresponding to these differential and difference equations. The project will provide research training opportunities for graduate students.In more detail, the project stems from transcendental construction of quantum groups from the Stokes data of the dynamical Knizhnik-Zamolodchikov equations for the corresponding Lie algebra due to the PI. The first component will extending the construction to numerical values of the deformation parameter, in particular to roots of unity, and to the difference setting. The second component will establish a Riemann-Hilbert correspondence for q-difference equations in several variables by defining an appropriate notion of regular singularities and capturing these by elliptic monodromy data, similar to the one-variable case treated by Birkhoff. The third component is concerned with the integrable systems arising from the Casimir connection, and their parametrisation in terms of sheets of the corresponding Lie algebra. The results of the project will have application in the study of Stokes phenomena, quantum integrable systems and geometric representation theory.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.