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，这是一种系统的方法，通过关注构成几何对象的整个过程(函子)来研究基本的几何对象，特别是各种函数的空间(例如嵌入)。它允许以一种层次化的方式对对象进行系统的分层，从而揭示满足“多项式”局部到全局行为的不变量。由汤姆·古德威利和迈克尔·韦斯在80年代末的S和90年代初的S率先提出，直到最近，这些方法的广泛力量才变得明显。主要研究人员Arone、Ching、Dwyer、Kuhn、Lesh和Turchin都参与了这一领域的重要发现，这些发现的范围从关于球面的周期同伦的新结果到给出纽结空间的新模型。运算理论是为研究满足特定代数性质(结合性、交换性等)的运算系统而开发的另一种代数机器。直到某种受控变形。PIS目前的工作导致了一种新的观点，即函数式演算与更多研究的算术理论有着深刻的联系，并且人们可能能够使用等变同伦方法来衡量后者比前者简单得多。在拓扑学中，人们正在研究几何对象，从流形(曲线和曲面的高维版本)和纽结，在几何拓扑学的情况下，到连续函数和结构环的空间，直到变形，在代数拓扑学的情况下。人们通过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“全局”对象是由“局部”片段构建的，则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)研究函数式演算组织和构造这种不变量的方法，(B)将其与代数运算的非常重要的理论运算理论联系起来，以及(C)通过研讨会和会议将这些方法介绍给广泛的数学家。这个项目中研究的方法应该会对许多正在进行的和广泛感兴趣的数学问题提供新的见解，从算法的拓扑复杂性到表示理论到拓扑场论。

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.