Mathematical Sciences: Cohomological and Homotopical Methods in Mathematical Physics

9504871斯塔谢夫研究员的研究关注的是他在早期研究分类空间和有理同伦理论时开发的技术的应用，并将这些技术与物理学家特别介绍的发展相结合。目前他特别关注四类问题：(I)同伦结合微分分次代数和Lie类似，特别是当它们出现在弦场论和自旋n-代数中时；(Ii)约束哈密顿系统的约化的同调方面，包括经典和量子，体现在BRST形式主义和Batalin-Fradkin-Vilkovisky复形及其推广中；(Iii)拉格朗日和更一般的外微分系统的同调方面，体现在Batalin-Vilkovisky及其推广的反场形式中；(Iv)某些模空间的紧化的组合拓扑，尤其是与戏曲、结理论相关的更高门类。所有这些都涉及“高维代数”，一维图是不够的。尽管在更大、更抽象的概括性中定义了这种结构，但它们出现在数学物理中或受到数学物理启发的结构是本提案的重点。在过去十年左右的时间里，数学物理的某些领域的工作，特别是粒子和弦理论，越来越多地使用上同调技术。在某些情况下，物理学家独立地重新发现了研究人员在早期研究项目中发明或开发的工具；最近，计划好的相互作用和合作使物理学家意识到并因此利用了他发明的概念，例如强同伦李代数。这些技术在物理环境中的进一步发展已经开始对更纯粹的数学研究产生影响，例如在外部微分系统中。因此，正如经常发生的那样，这种跨学科活动已被证明是双向的，预计会有更多的互惠互利。***

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. ***