Manifolds and calculus of functors

[摘要]获奖者：dms -0708601首席研究员：Thomas G. goodwillie所有拟作的研究都以各种形式与微积分因子有关。首先，将同伦微积分的主要结果推广到自然和高度的普遍性。将使用几何语言来发展与常微分学的类比，以便一个合适的同局部范畴得到诸如正切和余切空间、张量场空间、连接、微分算子等。语言应该对阐明观点和提出新方向都有用。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中某种意义上仅次于稳定伪同位素（可称为亚稳态伪同位素）的部分。它的导数将用与稳定理论相关的Hochschild和循环同调方法类似的方法来研究；在流形中涉及到循环的地方，现在应该有二级图映射到流形中。除了找出循环同调的亚稳态类似外，还有一个更基本的问题，即寻求理解非流形项下的亚稳态现象，就像瓦尔德豪森k理论对稳定理论所做的那样，微积分可能会提供线索。有一个序列：(0)同伦理论，其中微积分导致树；(1)代数k理论，其中微积分导致圆；(2)亚稳态理论，其中微积分导致秩二图。需要解决的一件事是流形中所有这些图之间的相互作用。函子演算涉及抽象的极端，但它有其接地气的根源。它是一种组织原理，因其与牛顿和莱布尼茨的普通微积分相似而得名。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。通过微积分的一般定理，这个函数的性质可以引出一个数字的计算，这个定理起初看起来很神奇。这里的故事也是类似的：有时候，关于某个数学实体的事实——现在不是数字，而是某种几何对象——要得到最好的证明，需要把它放在一个背景中，这个对象是一个庞大的此类对象家族的一部分——一个函子——并使用一些更古老的魔法。函子微积分的一个应用领域是流形拓扑。另一个是同伦理论。amamifold是一种涉及多变量的数学系统。这些在科学和数学中无处不在。流形拓扑研究这些系统本身，将它们视为几何对象；一个有n个变量的系统被看作是一个n维对象。同伦理论研究流形和其他物体时，忽略了大量的信息，只考虑最粗糙的特征。这种观点上的改变是一个强有力的想法，因为这种提炼过程会导致概念的清晰和新的方法。同伦理论是研究流形的一个重要工具，也是拓扑学的一个分支。因此，随着岁月的流逝，几何学出人意料地、自然地成长起来。在函子演算中，从某种意义上讲，可以把所有同伦理论看作一个几何对象。

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.