Manifolds and calculus of functors

摘要奖：DMS-0708601主要研究者：托马斯G.所有提出的研究都与各种形式的函数演算有关。首先，将同伦演算的主要结果推广到自然和高度的一般性。 几何语言将被用来发展与普通微分学的类比，这样一个合适的同伦范畴就可以得到诸如切和余切空间、张量场空间、联络、微分算子等等。这种语言应该既有助于阐明思想，又有助于提出新的方向。第二，在一般情况下研究代数K-理论的导数。第三，流形理论的一部分，在某种意义上是在稳定的伪合痕（它可以被称为亚稳定的伪合痕）之后，将从微积分的角度进行探讨。它的衍生物将被研究的方法类似于Hochschild和cyclic homology的方法是relatedto稳定的理论;其中循环在一个流形涉及，现在应该有映射的秩二图到流形。除了计算出循环同调的亚稳态类似物之外，还有一个更基本的问题，即像瓦尔德豪森K理论对稳定理论所做的那样，寻求在非流形条件下理解亚稳态现象，而微积分可能对此提供线索。有一个序列：（0）同伦理论，其中演算导致树，（1）代数K-理论，wherecalculus导致圆，（2）亚稳态理论，wherecalculus导致秩二图。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相互作用。函子演算涉及极端的抽象，但它有实际的根源。这是一个组织原则命名为相似的普通微积分的牛顿和莱布尼茨。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，这个数字是一个巨大的数字家族的一部分--一个数字函数。函数的性质可以导致，由微积分的一般定理，在第一次似乎像魔术，以计算的数量。这里的故事是相似的：有时候，关于某个数学实体的事实--现在不是一个数字，而是某种几何对象--最好的证明方法是把它放在一个上下文中，这个对象是一个庞大的此类对象家族的一部分--一个函子--并使用一些最近的魔法。函子演算的一个应用领域是流形拓扑。另一个是同伦理论。流形是一种涉及多个变量的数学系统。这些在科学和数学中无处不在。流形拓扑学研究这样的系统，把它们当作几何对象;一个N变量系统被看作是一个N维对象。同伦理论研究流形和其他对象的观点，其中大量的信息被忽略，只留下最粗糙的功能考虑。这种观点上的转变是一个强有力的想法，因为这种升华的过程会导致概念的清晰和新的方法。 同伦理论除了是研究流形的重要工具外，它本身也是拓扑学的一个分支。因此，随着岁月的流逝，几何学令人惊讶而又自然地成长。在函子演算中，从某种意义上说，把同伦理论看作是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.