RUI: Structure and computations in motivic and chromatic homotopy

在这项拨款下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。自从F.Morel和V.Voevodsky提出了Motivic Homoty理论以来，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，对从数论到控制我们物理宇宙的定律都很重要)已经成为可能。同时，色同伦理论和拓扑模形式为将代数几何数据转化为拓扑学和同伦论提供了工具。这项研究将工作在这些领域的交叉点，使用代数几何和同伦理论来相互启发。通过将这一计划的一部分纳入里德学院的论文课程，这项研究将增强里德学生的教育体验，让不同的本科生接触到研究水平的数学。奥姆斯比提出了一个学习色性和基元同伦理论的四重计划。首先，在与J·海勒的进一步合作中，他将阐述伽罗瓦理论控制稳定动机同伦范畴中一个特别容易处理的部分的方式。第二，继续与M.Behrens、N.Stapleton和V.Stojanoska合作，他将使用双变量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域上的Motivic片谱序列的收敛性质，最终导致在稳定的Motivic同伦层中的计算。最后，他计划引入一种带有框架转移的谱预压缩理论，并研究它们与关于稳定动机同伦范畴的基本问题的关系。

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.