Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics

上同调物理学是指物理学的一部分， 主要是规范和场理论，其中各种 上同调技术的应用越来越广泛。 的 本项目涉及技术的应用 在研究者以前的分类研究中， 空间和理性同伦理论作为一个代数拓扑。 它 特别是针对四类问题：（1）同伦 结合微分分次代数和李类比， 特别是当它们出现在弦场论和自旋n- （2）拟Hopf代数与双代数的变形; (3)Zamolodcdhikov四面体方程和广义 分类空间;（4）约化的同调方面， 约束哈密顿系统，包括经典和量子，作为 体现在BRST形式主义和巴塔利亚-弗拉德金-维尔科维奇 复杂性及其泛化。 (1)（2）具有很强的 相似性;这项研究的一个主要目的是了解 这种相似性的根本原因。 (3)涉及“更高 维代数”，这是（1）中代数的类似物， (2)而是从一维的结构开始 图表不够。 虽然定义在更大和更多 抽象的一般性，如数学中出现的结构 物理学是这项工作的重点。 另一方面，一位著名的数学物理学家 惊叹于“数学的不合理的有效性。“这 项目致力于观察的一个特殊情况，即， 代数拓扑的不合理的有效性。 它开始 研究人员发表的一篇论文 三十年前，作为一个代数拓扑学家被提名为一个关键 在共形弦理论和相关问题中的作用。 “更高” 他在那里发展的H-空间的“序结合性”已经消失了 通过麦克莱恩手中的一系列发展， 斯坦、内曼和斯滕贝格;乔亚尔和斯特里特;最后 Drinfeld在1989年，他对 物理学和低维拓扑学。 斯塔谢夫已经成为 热情的参与者在随后的兴趣爆炸， 这个和相关领域。

Cohomological physics refers to that part of physics, primarily gauge and field theories, in which a variety of cohomological techniques are seeing increasing application. The instant project is concerned with application of techniques developed in the investigator's previous study of classifying spaces and rational homotopy theory as an algebraic topologist. It is particularly directed to four classes of problems: (1) homotopy associative differential graded algebras and Lie analogs, especially as they occur in string field theories and spin n- algebras; (2) quasi-Hopf algebras and deformations of bialgebras; (3) Zamolodcdhikov's tetrahedral equation and generalized classifying spaces; (4) the homological aspects of reduction of constrained Hamiltonian systems, both classical and quantum, as embodied in the BRST formalism and the Batalia-Fradkin-Vilkovisky complex and its generalizations. (1) and (2) bear a very strong resemblance; one major thrust of this research being to understand the underlying reason for this resemblance. (3) involves "higher dimensional algebra," which is the analog of the algebra in (1) and (2) but beginning with structures for which one-dimensional diagrams are inadequate. Although defined in greater and more abstract generality, such structures as occur in mathematical physics are the focus of this work. In another context, a noted mathematical physicist has marveled at the "unreasonable effectiveness of mathematics." This project is devoted to a special case of that observation, namely, the unreasonable effectiveness of algebraic topology. It begins with the discovery that a paper the investigator published almost thirty year ago as an algebraic topologist was slated for a key role in conformal string theory and related matters. The "higher order associativity" of H-spaces that he developed there has gone through a sequence of developments at the hands of MacLane; Brustein, Ne'eman, and Sternberg; Joyal and Street; and finally Drinfeld, who, in 1989, gave deep and ingenious applications to both physics and low dimensional topology. Stasheff has become an enthusiastic participant in the ensuing explosion of interest in this and related areas.