Mathematical Sciences: Manifolds and Homotopy Theory

9509744古德威利这个项目使用了几种‘函数演算’来研究微分同胚空间和光滑嵌入空间的同伦类型。这些技术利用多重连通性估计来用特定值来描述各种类型的连续函子；例如，拓扑空间的函子可以从它在高度连通空间的值恢复，或者流形的子空间的函子可以从它在零维子空间的值恢复，或者实内积空间的函子可以从它在高维空间的值恢复。该项目将提炼和结合这些技术，并将它们应用于差分拓扑学中的各种问题。上面提到的每个“函数式演算”之所以被称为“函数式演算”，是因为它们与牛顿和莱布尼茨的普通“函数式演算”并不完全是想象中的相似之处。有时，把一个关于数字的事实放在一个上下文中来证明它是最好的，其中一个数字是一个巨大的数字家族的一部分--一个数字函数。然后，通过微积分的一般定理，函数的性质在第一次遇到它们时可能看起来有点神奇，从而导致数字的计算。这里就是这样：有时，通过将某个几何定义的对象放在这样的对象的一个庞大家族--函数器--的上下文中，并使用某种更现代的魔法来证明它，就是最好的证明。这个类比可能传达了一些研究的味道；内容更难传达，因为大多数正在讨论的几何对象只是通过相当长的抽象概念链与日常现实联系在一起的。(尽管有类似的长链抽象概念的介入，但人们发现，物理学家研究的对象预测真实物理现象的精度惊人。)*

9509744 Goodwillie This project uses several kinds of 'functorial calculus' to investigate the homotopy types of spaces of diffeomorphisms and spaces of smooth embeddings. These techniques exploit multirelative connectivity estimates to describe continuous functors of various types in terms of special values; for example, a functor of topological spaces might be recovered from its values at highly connected spaces, or a functor of subspaces of a manifold, from its values at zero-dimensional subspaces, or a functor of real inner product spaces, from its values at high-dimensional spaces. The project is to refine and combine these techniques and to apply them to various questions in differential topology. Each 'functorial calculus' mentioned above is so called because of a not-entirely-fanciful resemblance to the ordinary 'functional calculus' of Newton and Leibniz. Sometimes a fact about numbers is best proved by placing it in a context where a number is part of a huge family of numbers -- a numerical function. Properties of the function then lead, by general theorems of calculus that may seem a bit magical on first encountering them, to a computation of the number. So it is here: sometimes a fact about some geometrically defined object is best proved by placing it in a context where the object is part of a huge family of such objects -- a functor -- and using some magic of a more modern kind. This analogy may convey something of the flavor of the research; the content is harder to convey, because most of the 'geometric' objects in question are connected to everyday reality only by rather long chains of abstract ideas. (Despite the intervention of similar long chains of abstract ideas, however, one finds that the 'objects' physicists study predict real physical phenomena with uncanny precision.) ***