Calculus of Functors, Operads, and Manifolds

函子、运算和流形的微积分

基本信息

  • 批准号:
    0605073
  • 负责人:
  • 金额:
    $ 11.13万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2006
  • 资助国家:
    美国
  • 起止时间:
    2006-07-01 至 2010-06-30
  • 项目状态:
    已结题

项目摘要

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.
Arone对函子演算、操作数和几何拓扑之间的相互作用很感兴趣。他正在研究光滑嵌入空间的泰勒近似,他已经找到了树空间和图空间中这些近似的公式。他还打算表明(在与P. Lambrechts和I. Volic的联合工作中),小立方体运算的形式意味着对流形嵌入到欧几里得空间的空间的相当强的理性分裂结果。这些分裂结果将结理论的一些主要定理(如Vassiliev谱序列的坍缩)推广到一般嵌入空间。从长远来看,Arone希望将这项工作扩展到流形的微分同态研究。粗略地说,微分同态空间的一阶导数由Waldhausen的a -理论给出,或者由拓扑循环同调(TC)给出,Arone希望找到TC的高阶类似物,对应于高阶导数。圆群在TC定义中所扮演的角色在更高的程度上应该由一种固定同伦类型的图类来扮演,这种图类可能与研究自由群的自同构群的图的外空间有关。另一方面,Arone对进一步发展泛函演算的一般理论也很感兴趣。例如,他有兴趣发展一个多项式函子理论来推广古德威利的齐次函子理论。阿隆对这一主题的理解得到了秦先生工作的推动,他的工作明确了歌剧与这个问题的相关性。Arone对将函子演算应用于主流同伦理论也很感兴趣。在最近与K. Lesh合作的一项工作中,他发现了复k理论的一种新的过滤,以及这种过滤与函子演算之间的一种相当惊人的关系。这篇论文以一系列关于这种关系的确切性质的猜测结束,他想继续研究这些猜测。Arone认为,这一提议将为当前数学研究的活跃领域带来新的亮点,并将带来重要的新见解。本研究应将几何拓扑学的重要部分用代数拓扑学家所熟悉的语言表达出来,反过来将同伦理论的力量引入几何拓扑学。所提出的活动涉及代数拓扑、组合学、群上同调和几何拓扑等令人兴奋的思想相互作用。PI希望最终它能影响拓扑学以外领域数学家的思维。它还应该产生令人兴奋的新方法,向学生介绍拓扑学中的重要主题。它已经开始产生大量的数学博士论文。

项目成果

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Gregory Arone其他文献

The spectrum of excisive functors
切除函子的谱
  • DOI:
    10.1007/s00222-025-01338-9
  • 发表时间:
    2025-06-18
  • 期刊:
  • 影响因子:
    3.600
  • 作者:
    Gregory Arone;Tobias Barthel;Drew Heard;Beren Sanders
  • 通讯作者:
    Beren Sanders

Gregory Arone的其他文献

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{{ truncateString('Gregory Arone', 18)}}的其他基金

Mid-Atlantic Topology Conference
大西洋中部拓扑会议
  • 批准号:
    1535958
  • 财政年份:
    2015
  • 资助金额:
    $ 11.13万
  • 项目类别:
    Standard Grant
Calculus of Functors and Applications
函子微积分及其应用
  • 批准号:
    0307069
  • 财政年份:
    2003
  • 资助金额:
    $ 11.13万
  • 项目类别:
    Standard Grant
Calculus of Functors and Homotopy Theory
函子微积分与同伦论
  • 批准号:
    0196350
  • 财政年份:
    2000
  • 资助金额:
    $ 11.13万
  • 项目类别:
    Standard Grant
Calculus of Functors and Homotopy Theory
函子微积分与同伦论
  • 批准号:
    9971855
  • 财政年份:
    1999
  • 资助金额:
    $ 11.13万
  • 项目类别:
    Standard Grant
New Approaches to Global Homotopy Theory
全局同伦理论的新方法
  • 批准号:
    9704761
  • 财政年份:
    1997
  • 资助金额:
    $ 11.13万
  • 项目类别:
    Standard Grant

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RUI:操作数的 Koszul 对偶性和函子的微积分
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FRG: Collaborative Research: The Calculus of Functors and the Theory of Operads: Interactions and Applications
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运算和函子的演算
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运算和函子的演算
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