Mathematical Sciences: Cohomological and Homotopical Methods in Mathematical Physics

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9504871 Stasheff The investigator's research is concerned with application of techniques he developed earlier in his study of classifying spaces and rational homotopy theory and combining those techniques with developments introduced ad hoc by physicists. Currently he is particularly concerned with four classes of problems: (I) homotopy associative differential graded algebras and Lie analogs, particularly as they occur in string field theories and spin n-algebras, (II) the homological aspects of reduction of constrained Hamiltonian systems, both classical and quantum, as embodied in the BRST formalism and the Batalin-Fradkin-Vilkovisky complex and its generalizations, (III) 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, (IV) the combinatorial topology of compactifications of certain moduli spaces, particularly as related to operads, knot theory and higher categories. All of these involve "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 proposal. Over the last decade or so, work in some areas of mathematical physics, especially particle and string theory, has made increasing use of cohomological techniques. In some cases, physicists independently rediscovered tools the investigator had invented or developed in earlier research projects; more recently, planned interaction and collaboration has led to physicists' being aware of and hence making use of concepts he had invented, e.g., strong homotopy Lie algebras. Further development of these techniques within the physical context has begun to have an effect on more purely mathematical research, for example in exterior differential systems. Thus, as often happens, this in terdisciplinary activity has proved to be a two-way street, and further mutual benefits are anticipated. ***