Homotopy Theory, Geometry, and Arithmetic

代数拓扑学起源于回答几何形状具体问题的代数方法的发展。最初开发的最成功的工具之一是同调理论，它给出了关于形状的定性信息，如果形状被移动、扭曲或拉伸，这些信息保持不变。几十年来，人们致力于了解同源性，并制造出一系列新的、更通用的工具，这些工具共享相似的属性，但检测到不同类型的信息。开发了许多新的强大的工具；此外，还发现了与数论之间令人惊讶的联系。这项研究的重点是通过将最近的发展从数论提升到拓扑学来扩展我们对这种联系的理解，并将这种新的代数拓扑学机制应用于有关几何形状的具体问题，如纽结和链节。这一研究项目扩展了研究人员和合作者在构建结构环谱方面的工作，实现了与阿贝尔变种的特定模数的联系。通过色高度2考察了这些现象及其对稳定同伦理论中色过滤的影响。该项目的目的是通过对Picard模曲面的研究，在色高度3产生新的信息，并将这些结果应用于检测稳定同伦范畴中的高周期现象。此外，该项目将研究模理论和自同构型理论中的正性现象，以及它们与色同伦理论中的加法轨迹的联系。这项工作还旨在推广最近在构造与纽结相关的Khovanov谱方面的结果，并展示Khovanov关于与缠结相关的复形的工作提升到与缠结相关的稳定同伦型理论。最后，该项目将应用稳定等变同伦理论中的工作，特别是等变Tate对角线，来实现Heegaard-Floer同调中的同调水平构造。

Algebraic topology arose out of development of algebraic methods to answer concrete questions about geometric shapes. One of the most successful tools originally developed is homology theory, which gives qualitative information about shapes that remains unchanged if the shapes are moved, twisted, or stretched. Several decades were dedicated to understanding homology and producing a sequence of new, more general tools that shared similar properties but detected different types of information. Many new and powerful tools were developed; in addition, a surprising connection was discovered with number theory. This research focuses on expanding on our understanding of this connection by lifting recent developments from number theory to topology, and applying this new machinery of algebraic topology to concrete problems about geometric shapes such as knots and links. This research project extends work of the investigator and collaborators in constructing structured ring spectra realizing connections to specific moduli of abelian varieties. These phenomena and their impact on the chromatic filtration in stable homotopy theory have been examined up through chromatic height two. The project aims to produce new information at chromatic height three through the study of Picard modular surfaces and to apply these results to detecting higher periodic phenomena in the stable homotopy category. Further, the project will study positivity phenomena in the theories of modular and automorphic forms and their connection to the additive locus in chromatic homotopy theory. The work also aims to extend recent results in constructing Khovanov spectra associated to knots and to show Khovanov's work on complexes associated to tangles lifts to a theory of stable homotopy types associated to tangles. Finally, the project will apply work in stable equivariant homotopy theory, in particular the equivariant Tate diagonal, to realize homology-level constructions in Heegaard-Floer homology.