Homotopy Theory, Geometry, and Arithmetic

同伦理论、几何和算术

• 批准号: 1610408
• 负责人: Tyler Lawson
• 金额: $ 20.11万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1610408&HistoricalAwards=false
• 关键词: Homotopy; Theory; Geometry; Arithmetic

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.

代数拓扑学产生于代数方法的发展，用以回答有关几何形状的具体问题。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，如果形状被移动，扭曲或拉伸，这些信息将保持不变。几十年来，人们致力于理解同源性，并产生一系列新的、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。这项研究的重点是扩大我们的理解，这种连接，解除最近的发展，从数论拓扑，并应用这种新的机器代数拓扑的具体问题，如结和链接的几何形状。本研究延伸了研究者和合作者在构建结构环谱方面的工作，实现了与阿贝尔簇特定模的连接。这些现象及其对稳定同伦理论中色过滤的影响已经通过色高2得到了检验。该项目旨在通过研究Picard模块化曲面在色高3处产生新的信息，并将这些结果应用于检测稳定同伦类别中的更高周期性现象。此外，该项目将研究模形式和自守形式理论中的正性现象及其与色同伦理论中的加法轨迹的联系。这项工作的目的还在于扩大最近的结果在构建Khovanov光谱相关的结，并显示Khovanov的工作复杂的缠结升降机理论的稳定同伦类型相关的缠结。最后，该项目将应用稳定等变同伦理论的工作，特别是等变泰特对角线，以实现Heegaard-Floer同调中的同源级构造。