Functorial Calculus and Manifolds

9806981 Goodwillie这个项目主要是关于研究流形拓扑中超出代数K理论范围的那些方面的长期目标。这种方法是几种“函数式演算”的组合。一项任务是研究这些微积分之间的一些相互关系，以期加强一般方法。预计这将涉及到一些关于Poincare对偶复形的基础性工作。另一项长期的任务是研究可以被称为代数K理论的二阶模拟；它是由一种流形现象的二次稳定化而产生的。对于短期而言，由于循环同调与代数K-理论有关，因此需要构造一种与上述有关的二阶循环同调类似物。至少有一丝希望，这将产生四维流形的拓扑定义的不变量，类似于著名的解析定义的Seiberg-Witten不变量。“微积分”这个词之所以被使用，是因为它与普通的函数微积分有一个吸引人的类比。当然，有时解决数值问题的方法是将答案视为函数的值；借助强大的微积分设备，函数的性质可以导致对数字的计算。同样地，关于某些几何定义的对象的事实，有时最好的证明方法是将该对象视为函子的特定值，并使用某种函子演算的方法。这种类比可能会显示出作品的某种味道；其内容更难向非专家描述，因为大多数有问题的“几何”物体都通过相当长的抽象概念链与日常现实联系在一起。***

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. ***