Classical Methods in Motivic Homotopy Theory

动机同伦理论中的经典方法

• 批准号: 1906072
• 负责人: Haynes Miller
• 金额: $ 15.89万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906072&HistoricalAwards=false
• 关键词: Classical; Methods; Motivic; Homotopy; Theory

The subject of algebraic geometry concerns itself with the study of systems of polynomial equations. As it turns out, solving such systems exactly is, apart from a few very special cases, essentially impossible. We thus attempt to describe the solution sets more qualitatively. One classically fruitful approach to this is to consider the set of solutions as a topological space, an object sitting in some higher dimensional space. For instance, the equation x^2+y^2=1 describes a circle of radius 1. Topologists have invented invariants for qualitatively describing topological spaces, and we can just attempt to work out these invariants for the special spaces we are interested in. Taking this idea to its extreme, one arrives at a field called motivic homotopy theory. It provides novel ways for qualitatively describing solutions of systems of polynomial equations, and has in the past twenty years been successfully applied to several longstanding open problems in algebraic geometry. This project aims to deepen our understanding of motivic homotopy theory by exploring further not only its parallels with classical topology, but also what happens where these parallels break down.This project consists of several parts. One set of parts explores properties of motivic highly structured ring spectra, called normed spectra (introduced in collaboration with M. Hoyois). The Principal Investigator (P.I.) will (1) study power operations in normed spectra and deduce splitting results for normed spectra of positive characteristic and stability results for the motivic homology of symmetric groups and (2) construct further normed spectra, such as a normed spectrum structure on the motivic spectrum KO representing hermitian K-theory. Another set of parts explores the passage from unstable to stable motivic homotopy theory. Specifically the P.I. will (3) study the motivic Barratt-Priddy-Quillen map, and (4) study an unstable motivic homology Whitehead theorem. These goals will be achieved by using techniques from higher category theory, classical stable homotopy theory, algebraic topology and algebraic geometry. The results in parts (1) and (2) can be used to more effectively study algebraic varieties using cohomology theories; for example by exploiting the cohomology operations present in theories represented by normed spectra. Parts (3) and (4) are more useful for studying the motivic (stable) homotopy category itself, and hence the totality of all cohomology theories for algebraic varieties at once.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数几何学科涉及多项式方程组的研究。 事实证明，除了一些非常特殊的情况之外，精确地解决这样的系统基本上是不可能的。因此，我们尝试更定性地描述解决方案集。对此的一种经典有效方法是将解决方案集视为拓扑空间，即位于某个更高维度空间中的对象。 例如，方程 x^2+y^2=1 描述了一个半径为 1 的圆。拓扑学家发明了用于定性描述拓扑空间的不变量，我们可以尝试计算出我们感兴趣的特殊空间的这些不变量。将这一想法发挥到极致，我们就得到了一个称为动机同伦理论的领域。它提供了定性描述多项式方程组解的新颖方法，并且在过去二十年中已成功应用于代数几何中几个长期存在的开放问题。 该项目旨在通过进一步探索动机同伦理论与经典拓扑的相似之处，以及这些相似之处被打破时会发生什么，来加深我们对动机同伦理论的理解。该项目由几个部分组成。一组部分探索动机高度结构化环光谱的特性，称为赋范光谱（与 M. Hoyois 合作推出）。 首席研究员 (P.I.) 将 (1) 研究赋范谱中的幂运算，并推导出正特征赋范谱的分裂结果和对称群动机同调的稳定性结果，以及 (2) 构造进一步的赋范谱，例如代表埃尔米特 K 理论的动机谱 KO 上的赋范谱结构。 另一部分探讨了从不稳定动机同伦理论到稳定动机同伦理论的转变。具体来说，P.I.将 (3) 研究动机 Barratt-Priddy-Quillen 映射，以及 (4) 研究不稳定动机同调 Whitehead 定理。 这些目标将通过使用高范畴论、经典稳定同伦理论、代数拓扑和代数几何的技术来实现。 (1)和(2)部分的结果可以用来更有效地利用上同调理论研究代数簇；例如，通过利用赋范谱代表的理论中存在的上同调运算。第 (3) 和 (4) 部分对于研究动机（稳定）同伦范畴本身更有用，因此同时研究代数簇的所有上同调理论的整体。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。