Calculus of Functors and Homotopy Theory

小行星9971855 Arone感兴趣的是应用函子演算理论（由T。Goodwillie和M.韦斯）研究了球面空间和紧李群分类空间的不稳定同伦。 函子演算允许我们引入同伦群的某些“导数”。 如果一个人很好地理解这些导数，就可以用它们来恢复关于不稳定同伦理论的许多信息（这类似于这样一个事实，如果一个人知道一个普通函数在一点上的所有导数，那么他就可以通过泰勒级数恢复关于该函数的许多信息）。 导数实际上是稳定同伦论的对象，它们通常具有某些经典群的有趣对称性，例如对称群或正交群（被对称群的作用除对应于被n除！泰勒级数（Taylor Series） 因此，微积分允许一个约不稳定同伦理论的问题约equivariant稳定同伦理论。 这并没有产生一个有效的方法来计算同伦群（可能没有这样的方法），但它确实允许人们对同伦理论的整体结构做出有趣的定性陈述。 分析不同物体的衍生物，以及伴随的对称性，一直是Arone工作的重点。 这些衍生物本身似乎是非常有趣的对象，研究它们涉及到稳定同伦理论，等变拓扑和群上同调的方法相结合，以新鲜和意想不到的方式。 数学中的许多基本问题，特别是拓扑学中的许多基本问题，都具有以下形式：给定两个物体，描述它们之间可能的函数。 例如，人们可以要求描述两个拓扑空间之间的连续映射的空间，或者一个圆嵌入到某个物理空间中的空间（这被非正式地称为“结”空间）。 这些问题通常很难。 这是因为函数空间对源函数和目标函数的依赖关系非常复杂。非正式地说，可以说函数空间是一个复杂的二元函数。 事实证明，与普通微分学类似，人们可以为这种函数发展一种“泰勒级数”理论，并用它来近似复杂的空间，如用简单的空间来近似函数空间，就像用泰勒多项式近似普通函数一样。 这个非常惊人的想法是由T拜特提出的。好威利 Arone的大部分工作都集中在理解“泰勒多项式”和“导数”。 这被证明是一个迷人的和有益的旅程，这已经导致了有趣的新结果（代数）拓扑。 有些结果是这样一个普遍的性质，至少有一线希望，这项工作将产生影响数学以外的核心代数拓扑。

9971855Arone Arone is interested in applying the theory of calculus of functors(developed by T. Goodwillie and M. Weiss) to study the unstablehomotopy of such spaces as spheres and classifying spaces of compactLie groups. Calculus of functors allows one to introduce certain``derivatives'' of homotopy groups. If one understands thesederivatives well, one can use them to recover a lot of informationabout unstable homotopy theory (this is analogous to the fact that ifone knows all the derivatives of an ordinary function at a point, onecan often recover a lot of information about the function by means ofTaylor series). The derivatives are really objects of stable homotopytheory, and they normally come equipped with interesting symmetries ofcertain classical groups, such as the symmetric or orthogonal groups(division by the action of the symmetric group corresponds to divisionby n! in ordinary Taylor series). Thus calculus allows one to reducequestions about unstable homotopy theory to questions aboutequivariant stable homotopy theory. This does not yield an effectivemethod for calculating homotopy groups (there probably is no suchmethod), but it does allow one to make interesting qualitativestatements about the global structure of homotopy theory. Analyzingthe derivatives of different objects, together with the accompanyingsymmetries, has been the focus of much of Arone's work. Thesederivatives seem to be very interesting objects on their own, andstudying them involves combining methods from stable homotopy theory,equivariant topology, and group cohomology in fresh and unexpectedways. Many basic questions in mathematics, and in topology, in particular,are of the following form: given two objects, describe the possiblefunctions between them. For instance, one can ask to describe thespace of continuous maps between two topological spaces, or the spaceof embeddings of a circle into some physical space (this is informallyknown as the space of ``knots''). These questions are usually verydifficult. This is so because the dependence of the space offunctions on the source and the target is very complicated.Informally, one could say that the space of functions is a complicatedfunction of two variables. It turns out that, in analogy to ordinarydifferential calculus, one can develop a theory of ``Taylor series''of sorts for this kind of function and use it to approximatecomplicated spaces such as function spaces by simpler spaces, inmuch the same way as an ordinary function can be approximated by itsTaylor polynomials. This very striking idea was conceived byT. Goodwillie. Much of Arone's work has focused on understanding the``Taylor polynomials'' and the ``Derivatives'' in this context. Thishas proved to be a fascinating and rewarding journey, which hasalready led to interesting new results in (algebraic) topology. Someof the results are of such a universal nature that there is at least aslight hope that this work will have an impact on mathematics outsidehardcore algebraic topology.***