Abstracts

In this thesis we investigate the role of oddPoisson brackets in related areas of supergeometry. In particularwe study three different cases of their appearance: Courantalgebroids and their homotopy analogues, weak Poisson structuresand their relation to foliated manifolds, and the structure of oddPoisson manifolds and their modular class. In chapter 2 weintroduce the notion of a homotopy Courant algebroid, a subclass ofwhich is suggested to stand as the double objects toL-bialgebroids. We provide explicit formula for the higherhomotopy Dorfman brackets introduced in this case, and the higherrelations between these and the anchor maps. The homotopy Lodaystructure is investigated, and we begin a discussion of what otherconstructions in the theory of Courant algebroids can be carriedout in this homotopy setting. Chapter 3 is devoted to lifting aweak Poisson structure corresponding to a local foliation of asubmanifold to a weak Koszul bracket, and interpreting the resultsin terms of the cohomology of an associated differential. Thisbracket is shown to produce a bracket on co-exact differentialforms. In chapter 5 studies classes of second order differentialoperators acting on semidensities on an arbitrary supermanifold. Inparticular, when the supermanifold is odd Poisson, we given anexplicit description of the modular class of the odd Poissonmanifold, and provide the first non-trivial examples of such aclass. We also introduce the potential field of a general oddLaplacian, and discuss its relation to the geometry of the oddPoisson manifold and its status as a connection-likeobject.Advisors/Committee Members: KHUDAVERDYAN, HOVHANNES H, Voronov, Ted.