Calculus of Functors, Operads, and Manifolds

Arone感兴趣的是函子，运算和几何拓扑之间的相互作用演算。他正在调查泰勒近似空间的顺利嵌入，他发现这些公式近似空间的树木和空间的图形。他还打算展示（在与P. Lambrechts和I. Volic），小立方体运算符的形式意味着相当强的理性分裂结果的空间嵌入到一个流形的欧氏空间。这些分裂结果推广到一般的嵌入空间纽结理论的一些主要定理（例如，瓦西里耶夫谱序列的崩溃）。从长远来看，Arone希望将这项工作扩展到流形的超同态的研究。粗略地说，复同态空间的一阶导数由瓦尔德豪森的A-理论给出，或者由拓扑循环同调（TC）给出，而Arone希望找到TC的高阶类似物，对应于高阶导数。在TC的定义中，圈群所起的作用应该在更高的程度上由一类固定同伦类型的图所起，这可能与用于研究自由群自同构群的图的外层空间有关。在不同的静脉，Arone也有兴趣进一步发展的一般理论演算的函子。例如，他有兴趣发展一个理论的多项式函子，将推广古德威利的理论齐次函子。Arone对这一主题的理解得到了M.清，其中明确了相关的歌剧这个问题。Arone也对将函子演算应用于主流同伦理论感兴趣。在最近的一项工作中，K。莱什，他发现了一个新的过滤复杂的K理论和一个相当令人惊讶的关系之间的过滤和演算的函子。本文最后一系列的abstractures的精确性质的这种关系，他想追求这些abstractures.Arone认为，该提案将阐明新的光明活跃领域的当前研究数学，并将导致重要的新见解。拟议中的研究应该投下的一个重要组成部分的几何拓扑学的代数拓扑学家熟悉的条款，并反过来带来的权力同伦理论的几何拓扑。拟议的活动涉及到一个令人兴奋的相互作用的想法，从代数拓扑，组合学，群上同调和几何拓扑。PI希望它最终能影响到拓扑学以外领域的数学家的思维。它还应该产生令人兴奋的新方法，介绍拓扑学的重要课题给学生。它已经开始产生一些数学博士论文。

Arone is interested in the interplay between calculus of functors, operads and geometric topology. He is investigating the Taylor approximations to spaces of smooth embeddings, and he has found formulas for these approximations in terms of spaces of trees and spaces of graphs. He also intends to show (in a joint work with P. Lambrechts and I. Volic) that the formality of the little cubes operad implies quite strong rational splitting results for spaces of embeddings of a manifold into a Euclidean space. These splitting results generalize to general embedding spaces some major theorems of knot theory (e.g., the collapse of the Vassiliev spectral sequence). In longer term, Arone would like to extend this work to the study of diffeomorphisms of manifolds. Roughly speaking, the first derivative of the space of diffeomorphisms is given by Waldhausen's A-theory, or alternatively by topological cyclic homology (TC), and Arone would like to find the higher analogues of TC, corresponding to the higher derivatives. The role played by the circle group in the definition of TC should in higher degrees be played by a category of graphs a fixed homotopy type, perhaps related to the outer space of graphs that has been used to study the groups of automorphisms of a free group. In a different vein, Arone is also interested in developing further the general theory of calculus of functors. For instance, he is interested in developing a theory of polynomial functors that would generalize Goodwillie's theory of homogeneous functor. Arone's understanding of the subject was given a boost by the work of M. Ching, which made clear the relevance of operads to this question. Arone is also interested in applying calculus of functors to mainstream homotopy theory. In a recent work with K. Lesh, he discovered a new filtration of complex K-theory and a rather surprising relationship between this filtration and the calculus of functors. The paper ends with a series of conjectures about the precise nature of this relationship, and he would like to pursue these conjectures.Arone believes that the proposal will shed new light on active areas of current research in mathematics and will lead to important new insights. The proposed research should cast an important part of geometric topology in terms familiar to algebraic topologists, and conversely bring the power of homotopy theory to geometric topology. The proposed activity involves an exciting interplay of ideas from algebraic topology, combinatorics, group cohomology and geometric topology. The PI hopes that eventually it will impact the thinking of mathematicians from fields other than topology. It should also generate exciting new ways to introduce important topics in topology to students. It is already beginning to generate a number of PhD theses in mathematics.