CAREER: ARITHMETIC STRUCTURE OF HOMOTOPY THEORY

职业：同伦论的算术结构

• 批准号: 1452111
• 负责人: Mark Behrens
• 金额: $ 27.44万
• 依托单位: University of Notre Dame
• 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
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1452111&HistoricalAwards=false
• 关键词: CAREER; ARITHMETIC; STRUCTURE; HOMOTOPY; THEORY

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。