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.
Some earlier "best results": (1) de Rham theory for supermanifolds, discovery of new "variational" 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);
Current interests include analytic formulas for volumes of classical supermanifolds, microformal geometry and homotopy algebras, and super Darboux transformations.
You can find more information here.
I am in charge of the Manchester Geometry Seminar (together with my friend and collaborator Dr Hovhannes Khudaverdian, Prof. Nigel Ray, Dr Hendrik Suess and Dr James Montaldi).