Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics

上同调物理学指的是 物理学，主要是规范和其他场论，其中 各种上同调技术正在不断增加， 应用程序. Stasheff自己的研究关注的是 这些技术的应用，他早在 与他对空间和理性的分类研究有关 同伦理论 目前他特别关注 两类问题：I）的同调方面 约束哈密顿系统的简化，包括经典的 和量子，体现在BRST形式主义和 Batalin-Fradkin-Vilkovisky复形及其推广， II）同伦结合微分分次代数，因为它们 出现在弦场论和自旋n-代数中。 Henneaux和Stasheff将BFV复合物推广到 处理第一类的可还原约束，现在打算 将该结构应用于物理上有趣的示例。 关联同伦的类似物出现在字符串中 场理论，并导致Stasheff定义同伦 结合卷积代数和张量演算。 与 意大利物理学家科塔-拉穆西诺计划研究 这个同伦的物理意义和可能更高的 像他以前的作品一样的结构。 如此高阶 项在spin-n代数中是肯定存在的， 超过2. 他打算调查这些更高的秩序 将其转化为BRST公式。 这些应用程序中使用的许多技术来自 同调微扰理论Homological Perturbation Theory 这个项目包括 在理论上进一步发展这一机制 除了在理论物理学中的应用。

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. Stasheff's own research is concerned with application of those techniques he had earlier developed in relation to his study of classifying spaces and rational homotopy theory. Currently he is particularly concerned with two classes of problems: I) 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, and II) homotopy associative differential graded algebras as they occur in string field theories and spin n-algebras. Henneaux and Stasheff have generalized the BFV complex to handle reducible constraints of first class and now intend to apply the construct to physically interesting examples. Analogs of associating homotopies have appeared in string field theory and have led Stasheff to define a homotopy associative convolution algebra and tensor calculus. With the Italian physicist, Cotta-Ramusino, he plans to investigate the physical significance of this homotopy and possible higher order structures as in his previous work. Such higher order terms are definitely present in spin-n algebras for n bigger than 2. He intends to investigate these higher order structures by translating them into the BRST formulism. Many of the techniques used in these applications come from Homological Perturbation Theory. This project includes further development of that machinery at the theoretical level in addition to the applications in theoretical physics.