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年代对描述薄冰层的一维和二维统计力学模型的研究中发现的。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，该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种基本范式，描述了物理、化学、生物和经济系统的演化。它们引人注目的一个方面是它们可以展示斯托克斯现象：它们的解并不完全被用于求解它们的递归且通常是可编程的方法捕获。缺失的信息或斯托克斯数据可以被视为微分方程的隐藏对称性，因为它们涉及具有相同形式展开式的不同解。这个项目源于最近的一项发现，即量子群自然地产生于与经典对称有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对称性和量子对称性之间的桥梁。尤其令人感兴趣的是差分方程组的推广，它是微分方程的自然离散化，其斯托克斯数据除了单变量情况外并不能很好地理解。另一个重要的方向是研究与这些微分方程式和差分方程式对应的可积系统或运动常数。该项目将为研究生提供研究培训机会。更详细地说，该项目源于基于PI的相应李代数的动力学Knizhnik-Zamolodchikov方程的Stokes数据的超越构造量子群。第一个组件将把构造扩展到变形参数的数值，特别是单位根和差设置。第二部分将通过定义适当的正则奇点的概念并用椭圆单变量数据来捕捉这些正则奇点，从而建立多变量Q-差方程的Riemann-Hilbert对应，类似于Birkhoff处理的单变量情况。第三个分量涉及由Casimir联络产生的可积系统，以及它们在相应李代数的薄片上的参数化。该项目的成果将应用于斯托克斯现象、量子可积系统和几何表示理论的研究。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。

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.