Homotopy Theory, Geometry, and Arithmetic

代数拓扑学起源于回答几何形状的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源理论，它给出了关于形状的定性信息，这些信息在形状移动、扭曲或拉伸时保持不变。几十年来，人们致力于理解同源性，并产生了一系列新的、更通用的工具，这些工具具有相似的特性，但检测到不同类型的信息。许多新的和强大的工具被开发出来；此外，还发现了与数论的惊人联系。本研究的重点是扩展我们对这种联系的理解，从数论到拓扑学的最新发展，并将这种代数拓扑学的新机制应用于几何形状的具体问题，如结和链。本研究项目扩展了研究者和合作者在构建结构环谱方面的工作，实现了与阿贝尔变体特定模的连接。这些现象及其对稳定同伦理论中色光过滤的影响已经通过色光高度二得到了检验。该项目旨在通过对Picard模曲面的研究产生色高度为3的新信息，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，本项目将研究模和自同构形式理论中的正现象及其与色同伦理论中加性轨迹的联系。这项工作还旨在扩展最近在构造与结相关的Khovanov谱方面的结果，并将Khovanov关于与缠结相关的复合体的工作展示为与缠结相关的稳定同伦类型的理论。最后，本课题将运用稳定等变同伦理论，特别是等变Tate对角线中的工作，实现heegaard - flower同伦中的同调水平构造。

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.