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

FRG：协作研究：函子微积分和操作理论：交互和应用

• 批准号: 0967061
• 负责人: William Dwyer
• 金额: $ 7.79万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0967061&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Calculus; Functors

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.

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，特别是各种函数空间（例如嵌入）的系统方法，通过关注构造几何对象的整个过程（函子）。它允许以分层方式对对象进行系统分层，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年代末和90年代初首创，直到最近，这些方法的广泛力量才变得清晰起来。首席研究员Arone， Ching, Dwyer, Kuhn， Lesh和Turchin都参与了这一领域的重要发现，从关于球体周期同伦的新结果，到给出结空间的新模型。运算理论是另一种代数机器，用于研究满足特定代数性质（结合性、交换性等）直至某种控制变形的运算系统。pi目前的工作已经导致了一个新兴的观点，即函子演算与更深入研究的操作数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。应用的范围也在不断增长，在同域代数的更广泛的背景下放置微积分。在拓扑学中，人们正在研究几何对象，从几何拓扑学中的流形（曲线和曲面的高维版本）和结，到代数拓扑学中的连续函数和结构环的空间，再到变形。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”部分组成的，那么就会有一些过程允许人们尝试从局部不变量中计算出全局不变量。这个项目的目的是(a)研究用函子演算来组织和构造这些不变量的方法，(b)将其与运算理论，一个非常重要的代数运算理论联系起来，(c)通过研讨会和会议将这些方法带给广泛的数学家。在这个项目中研究的方法应该为许多正在进行和广泛感兴趣的数学主题提供新的见解，从算法的拓扑复杂性到表示理论到拓扑场理论。