CAREER: ARITHMETIC STRUCTURE OF HOMOTOPY THEORY

职业:同伦论的算术结构

基本信息

  • 批准号:
    1452111
  • 负责人:
  • 金额:
    $ 27.44万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2014
  • 资助国家:
    美国
  • 起止时间:
    2014-07-01 至 2017-06-30
  • 项目状态:
    已结题

项目摘要

The PI will investigate geometric invariants which associate automorphic forms to structured manifolds. The homotopy theoretic construction of these invariants is related to properties of holomorphic Eisenstein series on unitary groups. A geometric construction of these invariants will also be pursued, using higher dimensional field theories. These geometric invariants, when regarded as invariants of framed manifolds, give rise to invariants of elements of the stable homotopy groups of spheres. The arithmetic properties of such families of automorphic forms arising from periodic families in the stable homotopy groups of spheres will also be investigated. Similar analysis for unstable homotopy groups of spheres will be considered using Goodwillie calculus. The PI also proposes an educational program to identify and develop diverse undergraduate talent at MIT through ROUTE partnerships (Reading Outreach for Undergraduate Talent Exploration). These partnerships will pair MIT undergraduates who are interested in the mathematics major, but may not know what mathematicians do, with graduate student mentors to engage in semester long directed reading projects. These mentoring partnerships will be targeted to undergraduates who are members of underrepresented minority groups. The PI will solicit undergraduate research projects from outstanding students completing the ROUTE program, some of which will advance his own research agenda.The proposed research will advance our current understanding of geometry. It will also link this new understanding to physics, as the proposed research involves generalizations of string theory. As the proposed research involves using number theory to study geometry, it will associate new arithmetic structures to known geometric structures. The ROUTE partnerships will create a pathway to tap the diverse talent pool represented by the MIT undergraduate population, and will attract a more diverse collection of individuals to pursue the mathematics major.
PI将研究将自同构形与结构化流形联系起来的几何不变量。这些不变量的同伦理论构造与酉群上全纯Eisenstein级数的性质有关。这些不变量的几何构造也将使用更高维的场论来追求。当这些几何不变量被认为是框架流形的不变量时,就产生了球面稳定同伦群的元素的不变量。我们还将研究球面稳定同伦群中由周期族产生的这种自同构型族的算术性质。对于球面的不稳定同伦群,将使用古德威利演算进行类似的分析。PI还提出了一项教育计划,通过路径伙伴关系(本科人才探索阅读外展)来发现和培养麻省理工学院多样化的本科生人才。这些合作关系将把对数学专业感兴趣但可能不知道数学家是做什么的麻省理工学院本科生与研究生导师配对,参与为期一学期的定向阅读项目。这些指导伙伴关系将针对属于代表不足的少数群体的本科生。PI将向完成路线计划的优秀学生征集本科生研究项目,其中一些将推进他自己的研究议程。拟议的研究将促进我们目前对几何的理解。它还将把这种新的理解与物理学联系起来,因为拟议的研究涉及弦理论的泛化。由于拟议的研究涉及使用数论来研究几何,它将把新的算术结构与已知的几何结构联系起来。路径合作将开辟一条途径,挖掘以麻省理工学院本科生为代表的多样化人才库,并将吸引更多不同的个人攻读数学专业。

项目成果

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Mark Behrens其他文献

Topological Automorphic Forms
拓扑自守形式
  • DOI:
    10.1090/s0065-9266-09-00573-0
  • 发表时间:
    2007
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Mark Behrens;Tyler Lawson
  • 通讯作者:
    Tyler Lawson
On the top-weight rational cohomology of A g
关于 A g 的顶权有理上同调
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
    M. A. B. Randt;J. U. B. Ruce;M. E. C. Han;M. A. M. Elo;G. W. M. Oreland;C. O. W. Olfe;Mladen Bestvina;Mark Gross;Dan Abramovich;Arend Bayer;Mark Behrens;Jim Bryan;Mike Freedman;Colin Rourke;Roman Sauer
  • 通讯作者:
    Roman Sauer

Mark Behrens的其他文献

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{{ truncateString('Mark Behrens', 18)}}的其他基金

Conference: Midwest Topology Seminar
会议:中西部拓扑研讨会
  • 批准号:
    2341204
  • 财政年份:
    2024
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Standard Grant
Equivariant and Motivic Deformations of Stable Homotopy Theory
稳定同伦理论的等变和动机变形
  • 批准号:
    2005476
  • 财政年份:
    2020
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Standard Grant
Chromatic homotopy - stable and unstable
色同伦 - 稳定和不稳定
  • 批准号:
    1611786
  • 财政年份:
    2016
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Continuing Grant
CAREER: ARITHMETIC STRUCTURE OF HOMOTOPY THEORY
职业:同伦论的算术结构
  • 批准号:
    1050466
  • 财政年份:
    2011
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Continuing Grant
Conference Proposal: CURRENT AND CLASSICAL THEMES IN HOMOTOPY THEORY
会议提案:同伦理论的当前和经典主题
  • 批准号:
    0904858
  • 财政年份:
    2009
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Standard Grant
Local and global methods in homotopy theory
同伦理论中的局部和全局方法
  • 批准号:
    0605100
  • 财政年份:
    2006
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Continuing Grant
PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    0303415
  • 财政年份:
    2003
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Standard Grant

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稠密集中的算术结构
  • 批准号:
    2401117
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  • 批准号:
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  • 财政年份:
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Research of Learning by Building an Structure of Arithmetic Word Problem for Special Classroom
特殊课堂算术应用题构建学习研究
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  • 财政年份:
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论内存中算术事实的结构及其与数值幅度表示的相互作用
  • 批准号:
    394685337
  • 财政年份:
    2017
  • 资助金额:
    $ 27.44万
  • 项目类别:
    Research Grants
Arithmetic non-linear differential equations and Frobenius structure
算术非线性微分方程和 Frobenius 结构
  • 批准号:
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Self-inducing structure of arithmetic algorithms
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    Grant-in-Aid for Scientific Research (C)
Arithmetic Structure in the Integers and Higher-Order Fourier Analysis
整数的算术结构和高阶傅立叶分析
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    487479-2016
  • 财政年份:
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  • 项目类别:
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