RUI: Algebraic, Differential-Geometric, and Computational Aspects of Darboux Transformations in Classical and Super Settings

数学对称性是物体或结构的变换，使所研究的某些特征保持不变。 整个物理和工程科学中使用的对称性知识可以通过用更简单的物体代替复杂的物体来帮助理解它，或者可以帮助分离物理过程中保留的量，例如能量或动量。 该研究项目旨在发展作用于自然系统模型中普遍存在的微分方程的一类特定对称变换的理论。 该项目旨在将先前开发的代数方法与几何观点相结合，开发关于这些达布变换的新视角。 这将特别在“超级”设置中完成；这是指与超对称性相关的数学装置，超对称性是与基本粒子研究相关的理论概念。 研究人员将研究达布变换的代数方面，不同的变换如何相互组合，或者如何从基本块中构成它们，同时注意与坐标系选择无关的数量属性。该项目包括将研究结果应用到实用工具中的计划，例如用于求解微分方程的计算机软件。 该项目的另一个更广泛的影响将产生于为学生提供参与研究的机会。该项目的具体目标是使用先前开发的代数框架对一般形式算子的所有达布变换（DT）进行分类，该框架有助于证明二维薛定谔算子的 DT 因式分解以及发现一类新的可逆 DT。该项目旨在获得更多新型DT并开发精确求解算法（包括其计算实现）。研究人员打算在超几何环境中开发DT，这种设置与超对称偏微分方程的研究有关，将一维分类结果扩展到更高的维度。这项工作将考虑作用于几何对象的微分算子的DT，包括密度代数和微分形式。 初步研究表明，这将需要解决偏微分算子的新因式分解问题；该项目将探索使用正则化移动框架的微分不变量方法。子目标之一是分析作用于向量丛上的形式的算子。 研究人员还计划构建和研究由交织关系定义的 DT 的“通用流形”，并在 DT 和微分算子的“更高对称性”的最新概念之间建立联系。该项目将扩展基于 MAPLE 的软件包，以允许在超几何设置中使用具有参数系数的线性偏微分算子。

Mathematical symmetries are transformations of objects or structures that leave unchanged some characteristics under study. Knowledge of symmetries, used throughout the physical and engineering sciences, can aid understanding of a complicated object by replacing it with a simpler one or can help to isolate quantities that are preserved in physical processes, such as energy or momentum. This research project is aimed at developing the theory of a particular class of symmetry transformations that act on differential equations, which are ubiquitous in models of natural systems. The project aims to develop a new perspective on these Darboux transformations, combining a previously-developed algebraic approach with a geometric viewpoint. This will be done, in particular, in the "super" setting; this refers to the mathematical apparatus relevant for supersymmetry, a theoretical notion introduced in connection with study of elementary particles. The investigator will study algebraic aspects of Darboux transformations, how different transformations can be combined with each other or how they can be made from elementary blocks, with attention to properties of quantities independent of a choice of a coordinate system. The project includes plans to implement the results in practical tools such as computer software for solving differential equations. Another broader impact of the project will arise from providing opportunities for students to participate in the research.The specific goal of this project is to classify all Darboux transformations (DTs) for operators of general form using a previously-developed algebraic framework instrumental in the proof of factorization of DTs for two-dimensional Schrödinger operators and in discovery of a new large class of invertible DTs. The project aims to obtain more new types of DTs and to develop exact solution algorithms (including their computational implementation). The investigator intends to develop DTs in the supergeometric setting, which arises in connection with the study of supersymmetric partial differential equations, extending one-dimensional classification results to higher dimensions. The work will consider DTs for differential operators acting on geometric objects, including the algebra of densities and differential forms. Preliminary investigations show that this will require tackling new factorization problems for partial differential operators; the project will explore a differential invariants approach using regularized moving frames. One of the sub-goals is to analyze operators acting on forms on vector bundles. The investigator also plans to construct and study a "universal manifold" of DTs defined by the intertwining relation and to establish a connection between DTs and a recent notion of "higher symmetries" of differential operators. The project will extend a MAPLE-based package to allow work with linear partial differential operators with parametric coefficients in the supergeometric setting.