FRG: Collaborative Research: The Calculus of Functors and the Theory of Operads: Interactions and Applications

这个FRG项目的重点是函子演算，这是一种研究基本几何对象的系统方法，特别是各种函数的空间（例如嵌入），通过关注构造几何对象的整个过程（函子）。 它允许系统分层的对象在一个层次的方式，揭示了不变量，满足“多项式”的本地到全球的行为。 由Tom Goodwillie和Michael韦斯在80年代末和90年代初开创，直到最近，这些方法的广泛力量才变得清晰起来。主要研究员Arone，青，德怀尔，库恩，莱什，和Turchin都参与了重要的发现在这一领域，其中范围从新的结果有关周期同伦的领域，给新的模型空间的结。 运算理论是另一种代数机器，它被开发来研究满足特定代数性质（结合性，交换性等）的运算系统。直到某种受控的变形PI目前的工作已经导致了一种新兴的观点，即函子演算与更多研究的运算理论有着深刻的联系，并且人们可以使用等变同伦方法来衡量后者比前者简单多少。应用范围也随着微积分在同伦代数更广泛的背景下的位置而增长。在拓扑学中，人们正在研究从流形（曲线和曲面的高维版本）和结（在几何拓扑的情况下）到连续函数空间和结构环到变形（在代数拓扑的情况下）的几何对象。人们用代数不变量来研究这些东西。 这样的不变量需要是可计算的，这在实践中意味着，如果一个“全局”对象是由“局部”部分构建的，那么有一些过程允许人们尝试从局部不变量计算全局不变量。 这个项目的目的是（a）调查函子的演算方法组织和建设这样的不变量，（B）连接到运算理论，非常重要的理论代数运算，和（c）把这些方法，以广泛的数学家通过研讨会和会议。在这个项目中研究的方法应该给新的见解，许多数学主题的持续和广泛的兴趣，从拓扑复杂性的算法表示理论拓扑场论。

The focus of this FRG project is on the Calculus of Functors, a systematic method of studying fundamental geometric objects, particularly spaces of functions of various sorts (e.g. embeddings), through focusing on whole processes (functors) which construct the geometric objects. It allows for systematic stratification of objects in a hierarchical way that reveals invariants that satisfy `polynomial' local-to-global behavior. Pioneered by Tom Goodwillie and Michael Weiss in the late 80's and early 90's, it is only more recently that the broad power of these methods has been becoming clear. Principal Investigators Arone, Ching, Dwyer, Kuhn, Lesh, and Turchin have all been involved in important discoveries in this area, which range from new results about the periodic homotopy of spheres, to giving new models for spaces of knots. Operad Theory is another algebraic machine that has been developed to study systems of operations satisfying specified algebraic properties (associativity, commutativity, etc.) up to some sort of controlled deformation. The current work of the PIs has led to the emerging perspective that Functor Calculus has deep connections with the more studied Theory of Operads, and that one might be able to use equivariant homotopy methods to measure how much simpler the latter is than the the former. The range of application is also growing with the placement of Calculus within the broader context of Homotopical Algebra.In Topology, one is studying geometric objects ranging from manifolds (higher dimensional versions of curves and surfaces) and knots, in the case of Geometric Topology, to spaces of continuous functions and structured rings up to deformation, in the case of Algebraic Topology. One studies such things by means of algebraic invariants. Such invariants need to be computable, which in practice means that if a `global' object is built out of `local' pieces, there is some process that allows one to attempt to calculate the global invariant from the local invariants. The purposes of this project are to (a) investigate the Calculus of Functors method of organizing and constructing such invariants, (b) to connect this to Operad Theory, the very important theory of algebraic operations, and (c) to bring these methods to a broad spectrum of mathematicians through workshops and a conference. The methods studied in this project should give new insights into many mathematical topics of ongoing and wide interest, ranging from topological complexity of algorithms to representation theory to topological field theory.