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将研究将自动形式与结构化歧管联系起来的几何不变剂。 这些不变式的同型理论结构与单一群体上的全态爱森斯坦系列的特性有关。 使用较高维度的田地理论，也将追求这些不变性的几何结构。 这些几何不变剂被视为框形歧管的不变剂，从而产生了稳定的球体同型组的元素的不变性。 还将研究由稳定的球体中定期家族产生的自动形式家族的算术特性。对于不稳定的同型球体组的类似分析将使用Goodwillie演算来考虑。 PI还提出了一项教育计划，旨在通过路线合作伙伴关系在麻省理工学院识别和发展多样化的本科才能（阅读本科才能探索的外展）。 这些合作伙伴关系将使对数学专业感兴趣的麻省理工学院本科生与研究生导师与研究生导师一起参与长期导演的阅读项目。这些指导伙伴关系将针对代表人数不足的少数群体成员的大学生。 PI将从完成该路线计划的杰出学生那里征求本科研究项目，其中一些将推进他自己的研究议程。拟议的研究将提高我们当前对几何学的理解。 它还将将这种新理解与物理学联系起来，因为拟议的研究涉及弦理论的概括。 由于提出的研究涉及使用数字理论研究几何形状，因此它将将新的算术结构与已知的几何结构联系起来。 路线合作伙伴关系将创建一条途径，以利用以麻省理工学院本科生为代表的不同人才池，并将吸引更多样化的个人来追求数学专业。