CAREER: ARITHMETIC STRUCTURE OF HOMOTOPY THEORY

PI将研究将自守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Eisenstein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。 这些几何不变量，当被视为框架流形的不变量时，会产生球面的稳定同伦群的元素的不变量。 我们也将研究这些自守形式族的算术性质，这些自守形式族是由球面的稳定同伦群中的周期族产生的。球的不稳定同伦群的类似分析将考虑使用古德威利演算。 PI还提出了一个教育计划，以确定和发展多样化的本科人才在麻省理工学院通过路由伙伴关系（阅读拓展本科人才探索）。 这些合作伙伴关系将配对麻省理工学院的本科生谁是有兴趣的数学专业，但可能不知道数学家做什么，与研究生导师从事为期一学期的定向阅读项目。这些指导伙伴关系将针对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。拟议的研究将推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推广。 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。 ROUTE合作伙伴关系将创建一个途径，以利用麻省理工学院本科生人口为代表的多元化人才库，并将吸引更多元化的个人来追求数学专业。

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.