Differential Geometry, Topology, and Mathematical Physics.

    Much of what I do is related with supermanifolds and their applications. Broad areas of recent research can be described as bracket geometry (which includes odd and even Poisson structures, Lie algebroids and their generalizations, and homotopy algebras), geometry of differential operators, and integration on supermanifolds.

    Earlier "best results": (1) de Rham theory for supermanifolds, including discovery of a new "variational" construction of differential and links with Gelfand's general hypergeometric equations and integral geometry; (2) higher derived brackets, with applications to graded manifolds, homological vector fields, and Batalin-Vilkovisky geometry; (3) universal recurrence relations for super exterior powers, new formula for Berezinian as ratio of polynomial invariants, and applications to Buchstaber-Rees theory of n-homomorphisms (joint with H. Khudaverdian);

    My work also concerned quantization and Atiyah-Singer index theorem, quantum groups, and characteristic classes.

    Current interests include integration on supermanifolds, in particular analytic formulas for volumes of classical supermanifolds, microformal geometry and homotopy algebras, and super Darboux transformations, super Pluecker map with application to (not-fully-at-hand-yet) super cluster algebras, and other topics in supergeometry.





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Last modified: 11 (24) January 2023 Ted Voronov