Manifolds and calculus of functors

摘要奖：DMS-0708601首席研究员：托马斯·G·古德威利所有建议的研究都与微积分的各种形式有关。首先，将同伦演算的主要结果推广到自然的、高度的通用性。几何语言将被用来发展与常微分学的类比，以便合适的同伦范畴得到切线和余切空间、张量场空间、联络、微分算子等。这种语言应该对澄清想法和提出新的方向都有用。其次，将在一般情况下研究代数K-理论的导数。第三，从微积分的角度探讨流形理论在某种意义上紧随稳定伪同位素(可以称为亚稳态伪同位素)的部分。它的导数将通过类似于Hochschild的方法和与稳定理论相关的循环同调方法来研究；当涉及流形中的循环时，现在应该有等级2的图到流形的映射。除了求出环同调的亚稳类比之外，还有一个更根本的问题，就是寻求用非流形来理解亚稳现象，就像Waldhausen K理论对稳定理论所做的那样，而微积分可能对此有线索。有一个序列：(0)同伦理论，其中微积分通向树，(1)代数K-理论，其中微积分通向圆，以及(2)亚稳定理论，其中微积分导致等级2图。需要解决的一件事是流形中所有这些类型的图之间的相关类型的交互。函数演算涉及极端的抽象，但它有扎实的根源。这是一种组织原理，因为它与牛顿和莱布尼茨的普通微积分相似。有时候，一个关于数字的事实最好的证明方法是把它放在一个上下文中，这个数字是一个巨大的数字家族的一部分--一个数字函数。函数的性质可以通过微积分的一般定理引导到数的计算，这些定理乍看起来像是魔术。这里的故事是相似的：有时一个关于某个数学实体的事实--现在不是一个数字，而可能是某种几何对象--最好的证明是把它放在一个上下文中，这个对象是这类对象的一个大家族的一部分--一个函子--并使用一些最近的魔法。函数式演算的一个应用领域是流形拓扑。另一种是同伦理论。流形是一个涉及多个变量的数学系统。它们在科学和数学中无处不在。流形拓扑学是为了研究这类系统，将它们视为几何对象；N变量系统被视为N维对象。同伦理论从一个角度研究流形和其他对象，其中大量信息被忽略，只留下粗略的特征来考虑。这种观点的改变是一个强大的想法，因为这个蒸馏过程导致了概念的清晰和新的方法。同伦理论除了是流形研究的重要工具外，本身就是拓扑学的一个分支。因此，随着岁月的流逝，几何学自然地、令人惊讶地发展起来。在函子演算中，在某种意义上，将所有同伦理论视为一个几何对象是可能的。

AbstractAward: DMS-0708601Principal Investigator: Thomas G. GoodwillieAll of the proposed research is connected with the calculus offunctors in its various forms. First, the main results ofhomotopy calculus will be extended to a natural and high degreeof generality. Geometric language will be used to develop theanalogy with ordinary differential calculus, so that a suitablehomotopical category gets something like tangent and cotangentspaces, spaces of tensor fields, connections, differentialoperators, and so on. The language should be useful both forclarifying the ideas and for suggesting new directions. Second,the derivative of algebraic K-theory will be investigated in ageneral setting. Third, the part of manifold theory which in somesense comes next after stable pseudoisotopy (it could be calledmetastable pseudoisotopy) will be explored from the calculusviewpoint. Its derivatives will be studied by methods analogousto the Hochschild and cyclic homology methods that were relevantto the stable theory; where loops in a manifold were involved,now there should be maps of rank two graphs into the manifold. Inaddition to working out the metastable analogue of cyclichomology, there is the more fundamental question of seeking tounderstand the metastable phenomena in non-manifold terms asWaldhausen K-theory does for the stable theory, and calculus mayhold clues to this. There is a sequence: (0) homotopy theory,where calculus leads to trees, (1) algebraic K-theory, wherecalculus leads to circles, and (2) metastable theory, wherecalculus leads to rank two graphs. One thing to be worked out isthe relevant kinds of interactions between all these kinds ofgraphs in a manifold.Functor calculus involves extremes of abstraction, but it hasdown-to-earth roots. It is an organizing principle named forresemblance to the ordinary calculus of Newton andLeibniz. Sometimes a fact about numbers is best proved by placingit in a context where the number is part of a huge family ofnumbers -- a numerical function. Properties of the function canlead, by general theorems of calculus that at first seem likemagic, to a computation of the number. The story here is similar:sometimes a fact about some mathematical entity -- not a numbernow, but perhaps a geometric object of some kind -- is bestproved by placing it in a context where the object is part of ahuge family of such objects -- a functor -- and using some magicof more recent vintage. One area of application of functorcalculus is manifold topology. Another is homotopy theory. Amanifold is a kind of mathematical system involving manyvariables. These are ubiquitous in science andmathematics. Manifold topology studies such systems for their ownsake, treating them as geometric objects; an N-variable system isviewed as an N-dimensional object. Homotopy theory studiesmanifolds and other objects from a point of view in which a greatdeal of information is ignored, leaving only the coarsestfeatures to consider. This change in viewpoint is a powerfulidea, because this process of distillation leads to conceptualclarity and new methods. Besides being a key tool in the studyof manifolds, homotopy theory is a branch of topology in its ownright. Thus geometry grows, surprisingly and naturally, as theyears go by. In functor calculus it becomes possible in a senseto view all of homotopy theory as a geometric object.