Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics

上同调物理是指数学物理的一部分，主要是规范理论和其他场理论，各种上同调技术在其中的应用越来越广泛。Stasheff自己的研究是关于他早期在分类空间和理性同伦理论研究中开发的那些技术的应用。目前他特别关注两类问题：1)经典和量子约束哈密顿系统约简的同调方面，体现在BRST形式主义和Batalin-Fradkin-Vilkovisky复合体及其推广中；2)同伦关联微分梯度代数，出现在弦场理论和自旋n-代数中。Henneaux和Stasheff已经推广了BFV复合体来处理第一类的可约约束，现在打算将这个结构应用到物理上有趣的例子中。在弦场理论中出现了类似的关联同伦，并导致Stasheff定义了同伦关联卷积代数和张量微积分。他计划与意大利物理学家科塔-拉穆西诺（Cotta-Ramusino）一起，研究这种同伦的物理意义，以及他之前工作中可能出现的高阶结构。这种高阶项在n大于2的自旋n代数中肯定存在。他打算通过将这些高阶结构转化为BRST公式来研究它们。在这些应用中使用的许多技术来自于同调微扰理论。除了在理论物理中的应用外，该项目还包括在理论水平上进一步发展该机制。

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.