Luca Vitagliano

L_infinity algebras from multicontact geometry

I define higher versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher versions of pre-contact structures simply as distributions. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical sh Lie algebra to any (pre-)multicontact structure. Such sh Lie algebra is a higher version of the Jacobi brackets on contact manifolds. Since every partial differential equation (PDE) can be geometrically understood as a manifold with a distribution, then there is a (contact invariant) sh Lie algebra attached to any PDE.