Formal Groups, Structured Ring Spectra, and Stable Homotopy Theory

形式群、结构化环谱和稳定同伦理论

基本信息

  • 批准号:
    0706705
  • 负责人:
  • 金额:
    $ 28.66万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2007
  • 资助国家:
    美国
  • 起止时间:
    2007-07-01 至 2011-06-30
  • 项目状态:
    已结题

项目摘要

AbstractAward: DMS-0706705Principal Investigator: Paul G. GoerssThe chromatic picture of stable homotopy uses the algebraicgeometry of formal groups to organize and direct investigationsinto the deeper structure of computations and theory. Thisproject seeks to develop this point of view in two directions,one local and one global. The first, or local, direction is aninvestigation into K(n)-local homotopy theory in general and intothe K(2)-local sphere in particular. The second, or more global,direction, would be to make systematic our knowledge ofstructured ring spectra using stacks and the moduli stack offormal groups as the basic parameterizing device. In particular,a main part of the project is to continue work on the problem ofrealizing families of commutative ring spectra over the modulistack of formal groups. The spectrum of topological modular formsarises from taking the homotopy inverse limit of just such afamily and recent work of Lurie, Behrens, and Lawson had givennew examples. We can ask for systematic results along theselines, and we can ask for a thorough investigation into theexamples we have. An intriguing and novel feature of thesefamilies is that they use the theory of Barsotti-Tate groups tocombine information from formal groups of various heights.This project is in homotopy theory, which is a branch oftopology, a rather modern field that grew naturally out ofgeometry by studying phenomena that remain invariant undercontinuous transformations, rather than rigid (e.g.,angle-preserving) transformations. Of particular importance intopology are the continuous maps between large dimensionalspheres; under a suitable equivalence relation, this is the ringof stable homotopy groups of spheres. This notorious difficult tocalculate, or even to make conjectures about; therefore, in thepast few decades we have focused on trying to understandlarge-scale qualitative phenomena. In summary, this is the mainthrust of this project as well. It has been very fruitful todetect these phenomena using tools from other fields, especiallyalgebraic geometry. The transition from topology to geometry isdone using homology theories, which is a way of linearizingbehavior in topology. Simply sticking to one such theory is aradical process, however, and it loses too much data; therefore,we study families of such theories. Of particular importance isthe family parametrized by the stack of one-parameter formal Liegroups. The theory of stacks is vital here, as this allows us tostudy symmetries across continuous families of geometric objects-- especially when the self-symmetries can vary non-continuouslythroughout the family, as is most certainly the case here.
摘要奖:DMS-0706705主要研究者:Paul G. Goers稳定同伦的色图使用形式群的代数几何来组织和指导对计算和理论的更深层次结构的研究。本项目旨在从两个方向发展这一观点,一个是地方性的,一个是全球性的。 第一个方向,或者说局部方向,是对一般的K(n)-局部同伦理论和特别的K(2)-局部球面的研究。第二,或更全球性的,方向,将使我们的知识系统化的结构环光谱使用堆栈和模堆栈的形式群作为基本的参数化设备。 特别是,该项目的一个主要部分是继续工作的问题ofrealizing家庭的交换环谱上的modulistack正式集团。拓扑模形式的谱就是从这一类的同伦逆极限出发的,Lurie、Behrens和Lawson最近的工作给出了新的例子。我们可以要求沿着这些路线的系统性结果,我们可以要求对我们所拥有的例子进行彻底的调查。这些家族的一个有趣而新颖的特点是,他们使用Barsotti-Tate群的理论来结合来自各种高度的正式群的信息。这个项目是在同伦理论中,这是拓扑学的一个分支,一个相当现代的领域,通过研究在连续变换下保持不变的现象,而不是刚性(例如,保角)变换。在拓扑学中特别重要的是大维球面之间的连续映射;在适当的等价关系下,这是球面的稳定同伦群的环。这是出了名的难以计算,甚至难以解释;因此,在过去的几十年里,我们一直专注于试图理解大规模的定性现象。总之,这也是本项目的主旨。利用其他领域的工具,特别是代数几何工具来研究这些现象,已经取得了丰硕的成果。从拓扑学到几何学的过渡是使用同调理论完成的,这是拓扑学中线性化行为的一种方法。简单地坚持一个这样的理论是一个激进的过程,然而,它丢失了太多的数据;因此,我们研究这样的理论的家庭。 特别重要的是由单参数正式李群堆栈参数化的族。堆栈理论在这里是至关重要的,因为它使我们能够研究连续几何对象家族的对称性-特别是当自对称性在整个家族中可以不连续地变化时,就像这里的情况一样。

项目成果

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Paul Goerss其他文献

Realizing unstable injectives
  • DOI:
    10.1007/bf01163658
  • 发表时间:
    1987-06-01
  • 期刊:
  • 影响因子:
    1.000
  • 作者:
    Paul Goerss;Jean Lannes
  • 通讯作者:
    Jean Lannes

Paul Goerss的其他文献

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{{ truncateString('Paul Goerss', 18)}}的其他基金

Workshops in Spectral Methods in Algebra, Geometry, and Topology
代数、几何和拓扑谱方法研讨会
  • 批准号:
    2230159
  • 财政年份:
    2022
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Workshops: Homotopy Harnessing Higher Structures
研讨会:利用更高结构的同伦
  • 批准号:
    1833295
  • 财政年份:
    2018
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Conference on Derived Algebraic Geometry
派生代数几何会议
  • 批准号:
    1700795
  • 财政年份:
    2017
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Midwest Topology Seminar
中西部拓扑研讨会
  • 批准号:
    1747457
  • 财政年份:
    2017
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Midwest Topology Seminar, Spring 2014
中西部拓扑研讨会,2014 年春季
  • 批准号:
    1413786
  • 财政年份:
    2014
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Local and Global Chromatic Stable Homotopy Theory
局部和全局色稳定同伦理论
  • 批准号:
    1308916
  • 财政年份:
    2013
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Workshop in Equivariant, Chromatic, and Motivic Homotopy Theory
等变、半音和基元同伦理论研讨会
  • 批准号:
    1261225
  • 财政年份:
    2013
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
Chromatic Stable Homotopy Theory and Derived Algebraic Geometry
色稳定同伦理论及其派生代数几何
  • 批准号:
    1007007
  • 财政年份:
    2010
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Continuing Grant
Workshop on Homotopy theory and Derived Algebraic Geometry
同伦理论与派生代数几何研讨会
  • 批准号:
    1034873
  • 财政年份:
    2010
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant
The Topology and Geometry of Topological Field Theories
拓扑场论的拓扑和几何
  • 批准号:
    0852513
  • 财政年份:
    2009
  • 资助金额:
    $ 28.66万
  • 项目类别:
    Standard Grant

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