CAREER: ARITHMETIC STRUCTURE OF HOMOTOPY THEORY

PI将研究将自同构形式与结构流形相关联的几何不变量。这些不变量的同伦理论构造与酉群上全纯爱森斯坦级数的性质有关。这些不变量的几何结构也将采用高维场论。当把这些几何不变量看作框架流形的不变量时，就得到了稳定同伦群的球的元素的不变量。本文还研究了稳定同伦群中由周期族产生的自同构族的算术性质。用Goodwillie微积分对球面不稳定同伦群进行类似的分析。PI还提出了一项教育计划，通过ROUTE伙伴关系（本科人才探索阅读推广）在麻省理工学院识别和培养多样化的本科人才。这些伙伴关系将把对数学专业感兴趣，但可能不知道数学家在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性不足的少数群体的本科生。PI将从完成ROUTE项目的优秀学生中征集本科研究项目，其中一些将推进他自己的研究议程。这项提议的研究将促进我们目前对几何学的理解。它还将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推广。由于提出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。ROUTE的合作伙伴关系将为挖掘以麻省理工学院本科生为代表的多样化人才库创造一条途径，并将吸引更多不同的人来攻读数学专业。

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.