Mathematical Sciences: Manifolds and Homotopy Theory

数学科学：流形和同伦理论

• 批准号: 9509744
• 负责人: Thomas Goodwillie
• 金额: $ 8.99万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9509744&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Manifolds; Homotopy; Theory

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.) ***

小行星9509744 本计画利用几种函子演算来研究同伦型的同伦空间与光滑嵌入空间。 这些技术利用多相对连通性估计来描述各种类型的连续函子的特殊值;例如，拓扑空间的函子可以从其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其在零维子空间的值恢复，或者真实的内积空间的函子可以从其在高维空间的值恢复。 该项目是提炼和联合收割机这些技术，并将其应用到不同的问题，在微分拓扑。 上面提到的每一个“函子演算”之所以被称为“函子演算”，是因为它们与牛顿和莱布尼茨的普通“函子演算”有着不完全是想象出来的相似之处。 有时候，一个关于数字的事实最好的证明是把它放在一个上下文中，一个数字是一个巨大的数字家族的一部分-一个数字函数。 函数的性质，然后导致，由微积分的一般定理，可能似乎有点神奇，在第一次遇到他们，以计算的数量。 这里也是如此：有时候，关于某个几何定义的物体的事实，最好的证明是把它放在 一个上下文，对象是一个庞大的家庭的一部分，这样的对象-一个函子-和使用一些更现代的魔术。 这种类比可能传达出研究的某种味道;内容则更难传达，因为大多数所讨论的“几何”对象只是通过相当长的抽象概念链与日常现实相联系。 （然而，尽管有类似的一长串抽象概念的介入，人们发现物理学家研究的“对象”以不可思议的精确度预测了真实的物理现象。） ***