Mathematical Sciences: Manifolds and Homotopy Theory

这个项目使用了几种“泛函演算”来研究微分同态空间和光滑嵌入空间的同伦类型。这些技术利用多相对连通性估计用特殊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其在高连通空间中的值恢复，或者流形子空间的函子可以从其在零维子空间中的值恢复，或者实内积空间的函子可以从其在高维空间中的值恢复。本项目将对这些技术进行改进和组合，并将其应用于微分拓扑中的各种问题。上面提到的每一种“泛函演算”之所以被称为泛函演算，是因为它们与牛顿和莱布尼茨的普通“泛函演算”并不完全是奇思异想。有时候，关于数字的一个事实，最好的证明方法是把它放在一个庞大的数字家族——一个数值函数——的背景中。根据函数的性质，根据微积分的一般定理，第一次遇到这些定理时，可能会觉得有点神奇，然后可以计算出这个数。所以这里是这样的：有时候，关于一些几何上定义的物体的事实，最好的证明方法是把它放在这样一个庞大的物体家族的一部分——一个函子——并使用一些更现代的魔法。这个类比可能传达了一些研究的味道；内容很难传达，因为大多数“几何”对象都是通过相当长的抽象概念链与日常现实联系在一起的。（然而，尽管有类似的抽象概念的长链介入，人们发现物理学家研究的“对象”以不可思议的精度预测了真实的物理现象。）＊＊＊

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.) ***