Mathematical Sciences: Floer Homotopy, Kontsevich-Gromov- Witten Theory, and Quantum Cohomology

9504234 Morava凯勒流形的自由环空间的Floer同调在过去几年中一直是相当受关注的主题，部分原因是它定义了一个二维拓扑场论。最近，科恩、琼斯和西格尔定义了Floer同伦型的基本概念，它可以解释为这种拓扑场论到物理学家所称的二维拓扑引力理论的“普遍变形”。这种Floer同伦型是一个相当神秘的MU-代数谱。在这个项目中，研究人员为它构建了一个猜想模型，并探索了它从同伦理论的角度提供对弦物理学家的构造的理解的一些方法。整个课题对同伦理论和整体分析的未来有着深远的影响。十九世纪发展起来的力学理论是建立在“最小作用力原理”的基础上的：例如，一束光沿着使其飞行时间最短的路径运动。物理学家理查德·费曼(Richard Feynman)用一种关于所有可能路径的空间的积分理论重新解释了这些想法；他的想法现在是我们理解量子力学的基础。不幸的是，这种费曼路径积分的理论从来没有变得严谨；事实上，现在人们知道，任何对经典积分理论的幼稚概括都不能为现代物理学中出现的积分形成足够的基础。然而，在几何学中，最近变得清楚的是，费曼积分理论中的思想可以用来解决纯数学的经典问题，而且有证据表明，许多几何问题在某种意义上已经“驯服”到可以严格地建立起费曼积分理论的类比。虽然这些几何试题在许多方面受到限制，但它们为理解与物理直接相关的动态数学问题提供了非常清晰和极其重要的数据，它们是我们建立一致的费曼积分理论的最佳指南。在这个项目中，研究者用代数拓扑学的形式描绘了一类拓扑场论的猜想描述，这类拓扑场论是由费曼积分技术应用于复杂流形的几何而产生的。***

9504234 Morava The Floer homology of the free loopspace of a Kaehler manifold has been the subject of considerable attention in the last few years, in part because it defines a two-dimensional topological field theory. Recently Cohen, Jones, and Segal have defined an underlying notion of Floer homotopy type, which can be interpreted as the `universal deformation' of this topological field theory to what physicists call a theory of two-dimensional topological gravity. This Floer homotopy type is a rather mysterious MU-algebra spectrum. In this project the investigator constructs a conjectural model for it, and he probes some of the ways in which it provides an understanding of the constructions of string physicists from a homotopy-theoretical point of view. The whole subject has profound implications for the future of homotopy theory and global analysis. The theory of mechanics developed in the ninteenth century was based on 'principles of least action': a ray of light, for example, follows the path which minimizes its time of flight. The physicist Richard Feynman reinterpreted these ideas in terms of a theory of integration over the space of all possible paths; his ideas are now fundamental to our understanding of quantum mechanics. Unfortunately, the theory of such Feynman path integrals has never been made rigorous; indeed, it is now known that no naive generalization of the classical theory of integration can form an adequate basis for the integrals which arise in modern physics. In geometry, however, it has become clear recently that ideas from the theory of Feynman integrals can be used to solve classical problems of pure mathematics, and there is evidence that many geometric problems are in some sense 'tame' enough so that an analogue of the theory of Feynman integrals can be established rigorously. Although restricted in many ways, these geometrical test questions provide very clear and extremely important data for understanding the 'dyna mical' problems of direct interest to physics, and they are our best guide to a consistent theory of Feynman integrals. In this project the investigator sketches a conjectural description, in terms of algebraic topology, for a rather lapge class of topological field theories, which arise from the application of Feynman integral techniques to the geometry of complex manifolds. ***