代数拓扑中的代数几何与数论方法

This project aims to further the study of algebraic topology by employing methods and machinery developed in algebraic geometry and number theory. The PI proposes the following three directions of research: local structures of moduli spaces for algebraic varieties and formulas for power operations; deformation theory of formal groups and existence of highly-structured orientations; actions of differential operators on quasimodular forms and properties of Rezk's logarithms...The main focus of the first direction is cohomology operations, one of the central tools in algebraic topology. The goal is to obtain explicit formulas for power operations in chromatic homotopy theory, which will provide researchers in the field with new computational tools, generating potential applications to certain problems in algebra and geometry. An input from algebraic geometry will be to generalize the PI's computations for local moduli of elliptic curves to higher-dimensional Abelian varieties...The second direction has its background in classifications and invariants of topological manifolds. The goal is to show the existence of highly-structured orientations on topological modular forms and related cohomology theories, which will produce new invariants. To this end, from the perspective of algebraic geometry and number theory, the PI will study the deformation theory of one-dimensional formal groups. In particular, he will investigate a conjecture of Rezk on connections between Ando coordinates of universal deformations and Coleman norms from class field theory...Third, the PI hopes to further the study of differential operators acting on quasimodular forms, in order to understand certain properties of Rezk's logarithms and related cohomology operations.

本项目旨在运用代数几何、数论中发展起来的方法和工具推进代数拓扑的研究，拟包括三个方面：代数簇模空间的局部结构与幂运算的表达式；形式群的形变理论与高度结构化定向的存在性；拟模形式上的微分算子作用与Rezk对数的性质。第一个方面是关于上同调运算这项代数拓扑核心工具的研究。目标是给出色展同伦论中幂运算的显式表达，为同行们提供新的计算工具，并应用于解决几何与代数中的某些问题。需要的代数几何方法是将申请人对椭圆曲线局部模空间的计算推广到高维Abel簇。第二方面的背景是拓扑流形的分类和不变量，其目标是证明拓扑模形式及相关上同调理论高度结构化定向的存在性，以输送新的不变量。为此，从代数几何与数论的角度，研究一维形式群的形变理论，特别是Rezk关于万有形变上Ando坐标与类域论中Coleman范数之联系的猜想。第三，继续申请人关于拟模形式上微分算子作用的研究，以揭示相关上同调运算特别是Rezk对数的性质。

本项目的研究工作围绕广义上同调理论和上同调运算，在椭圆上同调幂运算的结构与计算方面取得了突出成果，包括椭圆曲线模空间的局部方程与椭圆上同调幂运算的显式表达及其应用、Hecke 算子的作用与 Rezk 对数运算的核、形式群的形变理论与流形的高度结构化配边不变量。其工作具有鲜明的学科交叉特征，用代数几何、数论的方法解决代数拓扑学中的问题，将代数簇模空间的局部结构、模形式的 Fourier 展开、Hecke 算子等部件装配并开发为具体的代数拓扑计算工具，以使人们的认识更进一步。