This research is concerned with application of techniques Stasheff developed earlier in his study of classifying spaces and rational homotopy theory and involves his combining of those techniques with developments introduced ad hoc by physicists. The current project is concerned particularly with three inter-related classes of problems and aims at elucidating their intrinsic structure as well as computation of significant applications: (I) homotopy associative differential graded algebras and Lie and commutative analogs, particularly as they occur in various physical field theories, (II) the homological aspects of Lagrangian and more general exterior differential systems, both classical and quantum, as embodied in the anti-field formalism of Batalin-Vilkovisky and its generalizations, (III) deformation theory as giving rise to higher homotopy algebra and as applied to physical systems. Specific applications are projected to the problems of higher spin particles and of mixed open-closed string field theory. Cohomological physics refers to that part of mathematical physics, primarily gauge and other field theories, in which a variety of cohomological techniques are seeing increasing application. Recently, further development of these techniques within the physical context has begun to have an effect on more purely mathematical research, for example, providing new applications of the existing theory of ``higher dimensional algebra'' for which 1-dimensional diagrams are inadequate. Although defined in greater and more abstract generality, such structures, as they occur in or are inspired by mathematical physics, are the focus of this research. The results should aid in deeper understanding of the mathematical structures essential to the physics (especially of higher spin particles and of mixed open-closed string field theory) and of the inter-relation of physical and mathematical concepts. The results should also be of independent mathematical importance. ***
