Calculus of Functors and Applications

DMS-0307069 Gregory Z.Arone这个项目的主要目标是将函子演算，特别是M.Weiss发展的“正交演算”版本，应用到嵌入空间的研究中。更详细地说，设M，N是光滑流形。PI希望通过取这些流形中的一个或两个与一般欧氏空间的笛卡儿积，并研究所获得的欧氏空间的函子的性质，来研究嵌入Emb(M，N)空间。微积分的一般理论把一个“导数”序列和一个“泰勒塔”联系在一起，“泰勒塔”是一个由多项式函子逼近的序列。对嵌入函子导数的研究使人们考虑了一系列非常美丽的拓扑结构，有些是经典的，有些是新的。我们遇到的经典结构包括Fulton-McPherson紧化、划分偏序集和树空间(推广)。PI希望这个项目将对嵌入空间的拓扑和流形的自同构产生重要的新见解。流形是数学的基本研究对象之一。流形有不同的维度。一维流形是曲线，二维流形是曲面，高维流形是这些概念的适当扩展。关于流形的一个基本问题是：给定流形M，M的可能对称(微分同胚)是什么？正是T.Goodwillie提出了一个非常引人注目的想法，即人们应该系统地研究对称空间(或我们想研究的任何东西)是如何变化的，因为一个人改变了流形，而不是一次处理一个流形。这导致了一种类似于经典微积分的理论，在经典微积分中，函数是通过其导数、泰勒多项式等来研究的。这个思想为研究流形(和其他数学感兴趣的对象，特别是拓扑)提供了一个强大而美丽的框架，包含了相当多的经典技术，并引导人们在拓扑学中发现美丽的新结构。

DMS-0307069Gregory Z. AroneThe main goal of this project is to apply calculus of functors, especially the"orthogonal calculus" version developed by M. Weiss, to the study of spaces ofembeddings. In more detail, let M, N be smooth manifolds. The PI would like tostudy the space of embeddings Emb(M,N) by taking the cartesian product of one ortwo of these manifolds with a generic Euclidean space, and investigating theproperties of the obtained functor of the Eucliedan space. The general theory ofcalculus associates with such a functor a sequence of "derivatives", where then-th derivative is a spectrum with an action of the orthogonal group O(n), and a"Taylor tower" - a sequence of approximations by polynomial functors. The studyof the derivatives of the embedding functor leads one to consider a ratherbeautiful array of topological constructions, some classical, some new. Theclassical constructions that one encounters include (a generalization of) theFulton-McPherson compactification, partition posets and spaces of trees. The PIhopes that this project will yield significant new insights into the topology ofspaces of embeddings and automorphisms of manifolds.Manifolds are among the basic objects of study in mathematics. Manifolds come indifferent dimensions. One dimensional manifolds are curves, two dimensionalmanifolds are surfaces, and high dimensional manifolds are suitable extensionsof these concepts. One of the basic questions about manifolds is: given amanifold M, what are the possible symmetries (diffeomorphisms) of M? It was T.Goodwillie who came up with the very striking idea that rather than approachsuch questions "one manifold at at a time", one should study systematically howthe space of symmetries (or whatever it is we want to study) changes, as onevaries the manifold. This results in a theory analogous to the classicaldifferential calculus, where functions are studied via their derivatives, Taylorpolynomials and so forth. This idea provides one with a powerful and beautiful framework for studying manifolds (and other objects of interest in mathematics, especially topology), subsumes a fair amount of classical techniques, and leads one to discover beautiful new constructions in topology.