This is a joint project with E. Shemyakova of the University of Toledo (Ohio). In brief, a Darboux transformation (DT) is a non-group type symmetry of differential operators. It maps an operator to another operator "of the same form" (e.g., with the same principal symbol) together with a linear transformation of the kernels or general eigenspaces. Such transformations originated in the works of Euler, Laplace, Darboux and other classics and were rediscovered in connection with integrable systems in 1970s. An algebraic description of Darboux transformations is via certain intertwining relations. A lot is known about particular examples of Darboux transformations and their applications, but less is known about general theory.
The project is devoted to developing general theory of Darboux transformations, including algebraic, geometric and computational aspects. In particular, our goal is to study Darboux transformations in the context of supergeometry. As the first step, we have obtained a complete classification of DTs for arbitrary nondegenerate differential operators on the superline (dimension 1|1). This study forced us on the way to advance certain aspects of superalgebra and supergeometry (e.g. expansions of Berezinians). Another attractive direction is to include a construction that arose in Batalin-Vilkovisky geometry, namely, differential operators on the algebra of densities (introduced in my earlier works with H. Khudaverdian). This is in progress, we have started to study factorizations for such operators, where non-trivial features already arise.
Most recently this project has developed towards a construction of a superanalog of the classical Pluecker embedding.
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Last modified: 19 September (2 October) 2019 Ted Voronov