Functorial Calculus and Manifolds

小行星9806981 这个项目主要关注的长期目标是调查那些方面的拓扑流形的谎言超出了代数K理论。 该方法是一种组合 有几种“函子演算"。“一项任务是调查一些 这些结石之间的相互关系，着眼于加强一般方法。 预计这将涉及一些基础性工作 关于庞加莱对偶复合体 另一项长期任务是研究所谓的代数K理论的二阶类似物;它是由流形现象的二次稳定化引起的。 对于短程有工作的建设一种二阶类似的循环同调，应与上述有关的循环同调是有关代数K理论。 至少有一点希望，这将产生拓扑定义的四维流形的不变量，类似于著名的解析定义的塞伯格-威滕不变量。 使用“微积分”一词是因为它与普通的函数微积分有一个吸引人的类比。 当然，有时解决一个数值问题的方法是把答案看作一个函数的值;借助于微积分的强大工具，函数的性质可以导致对数字的计算。 同样地，关于某个几何定义的对象的事实，有时最好通过将对象视为函子的特定值并使用某种函子演算的工具来证明。这种类比可能显示出作品的某种风格;内容很难向非专家描述，因为大多数所讨论的“几何”对象都通过相当长的抽象概念链与日常现实联系起来。 ***

9806981 Goodwillie This project is mainly concerned with the long-range goal of investigating those aspects of the topology of manifolds that lie beyond the reach of algebraic K-theory. The method is a combination of several kinds of "functor calculus." One task is to investigate some interrelationships between these calculi with an eye to strengthening the general method. This is expected to involve some foundational work on Poincare duality complexes. Another task for the long haul is to look at what may be called a second-order analogue of algebraic K-theory; it arises by a kind of quadratic stabilization of manifold phenomena. For the short haul there is the job of constructing a kind of second-order analogue of cyclic homology that should be related to the above as cyclic homology is related to algebraic K-theory. There is at least a slight hope that this would yield topologically defined invariants for four-manifolds that are akin to the well-known analytically defined Seiberg-Witten invariants. The word "calculus" is used because of an attractive analogy with the ordinary calculus of functions. Sometimes, of course, the way to settle a numerical question is to see the answer as a value of a function; by means of the powerful apparatus of calculus, properties of the function can lead to a computation of the number. Likewise, a fact about some geometrically defined object is sometimes best proved by viewing the object as a particular value of a functor and using the apparatus of some kind of functor calculus. This analogy may show something of the flavor of the work; the content is harder to describe to the non-expert, because most of the "geometric" objects in question are connected to everyday reality by rather long chains of abstract ideas. ***