Theodore Voronov (University of Manchester)
Most differentials arising in topology and homological algebra fall into one of the two classes: they are either simplicial or of Koszul/ de Rham type. About twenty five years ago, the speaker discovered another construction of a differential without any obvious alternation or exterior multiplication. It acts on Lagrangians of multidimensional surfaces and is expressed in terms of variational derivatives. (The identity d2=0 for it has a purely geometrical origin.) Combined with some other ideas, this helped to solve the problem of the Cartan -- de Rham complex for supermanifolds.
In this talk, I shall recall that construction of the variational differential and explain how it can be used in the inverse problem of calculus of variations: to find whether given functions can be the left-hand sides of the Euler--Lagrange equations for some functional. The classical Helmholtz criterion giving an answer to this question basically deals with a 1-form on an infinite-dimensional manifold of fields and checks whether such a form is closed or not. Our approach allows to consider 0-forms ("functions" or functionals) instead of 1-forms; to get an answer, one just needs to do a simpler check whether the variation of a certain functional is identically zero. I shall explain a few facts about forms on infinite-dimensional spaces.