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

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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.