RUI: Structure and computations in motivic and chromatic homotopy

RUI：动机和色同伦的结构和计算

• 批准号: 1406327
• 负责人: Kyle Ormsby
• 金额: $ 17.21万
• 依托单位: Reed College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406327&HistoricalAwards=false
• 关键词: RUI; Structure; computations; motivic; chromatic

The research conducted under this grant will study the interface between two fields of mathematics: homotopy theory and algebraic geometry. Since the introduction of motivic homotopy theory by F. Morel and V. Voevodsky, it has been possible to use homotopy theoretic methods to study schemes (rigid geometric objects controlled by algebraic equations, important to topics as far-ranging as the theory of numbers and the laws controlling our physical universe). Meanwhile chromatic homotopy theory and topological modular forms provide tools for transferring algebro-geometric data into topology and homotopy theory. This research will work at the intersection of these fields, using algebraic geometry and homotopy theory to inform each other. By incorporating portions of this program into the thesis curriculum at Reed College, this research will enhance the educational experience of Reed's students, exposing a diverse population of undergraduates to research-level mathematics.Ormsby proposes a four-fold program for studying chromatic and motivic homotopy theory. First, furthering joint work with J. Heller, he will explicate the fashion in which Galois theory controls a particularly tractable part of the stable motivic homotopy category. Second, continuing joint work with M. Behrens, N. Stapleton, and V. Stojanoska, he will use bivariable modular forms and the Adams spectral sequence to study cooperations in topological modular forms. Third, he plans to study the convergence properties of the motivic slice spectral sequence over infinite-cohomological dimension fields, ultimately leading to computations in stable motivic homotopy sheaves. Finally, he plans to introduce a theory of spectral presheaves with framed transfers and study their relation to foundational questions about the stable motivic homotopy category.

根据这项补助金进行的研究将研究两个数学领域之间的接口：同伦理论和代数几何。 自F. Morel和V.Voevodsky，已经可以使用同伦理论方法来研究方案（由代数方程控制的刚性几何对象，对数论和控制我们物理宇宙的定律等广泛主题很重要）。 色同伦理论和拓扑模形式为代数几何数据向拓扑同伦理论的转化提供了工具。 这项研究将在这些领域的交叉点工作，使用代数几何和同伦理论相互通报。 通过将该计划的一部分纳入里德学院的论文课程，这项研究将提高里德的学生的教育经验，使不同人口的本科生接触到研究水平的mathematics.Ormsby提出了一个学习色同伦理论和动机同伦理论的四重计划。 首先，进一步与J. Heller合作，他将解释伽罗瓦理论控制稳定动机同伦范畴中特别容易处理的部分的方式。 第二，继续与M. Behrens，N.斯台普顿和V.Stojanoska，他将使用二元模形式和亚当斯谱序列来研究拓扑模形式中的合作。 第三，他计划研究无限上同调维数域上的motivic切片谱序列的收敛性质，最终导致稳定motivic同伦层的计算。 最后，他计划介绍一个理论的频谱presheaves框架转让和研究他们的关系的基本问题的稳定motivic同伦范畴。