Classical Methods in Motivic Homotopy Theory

代数几何的主题涉及本身的研究系统的多项式方程。 事实证明，除了一些非常特殊的情况外，精确地解决这样的系统基本上是不可能的。因此，我们试图更定性地描述解集。一个经典的富有成效的方法是考虑作为一个拓扑空间，一个对象坐在一些高维空间的解决方案的集合。 例如，方程x^2+y^2=1描述了半径为1的圆。 拓扑学家已经发明了定性描述拓扑空间的不变量，我们可以尝试为我们感兴趣的特殊空间计算出这些不变量。把这个想法发挥到极致，我们就得到了一个叫做动机同伦理论的领域。它为定性描述多项式方程组的解提供了新的方法，并且在过去的二十年中已经成功地应用于代数几何中几个长期存在的开放问题。 这个项目旨在加深我们对动机同伦理论的理解，不仅进一步探索它与经典拓扑学的相似之处，而且探索这些相似之处被打破时会发生什么。这个项目由几个部分组成。一组部分探索motivic高度结构化的环光谱，称为赋范光谱（与M。Hoyois）。 主要研究者（P.I.）将（1）研究赋范谱中的幂运算，并推导出正特征的赋范谱的分裂结果和对称群的动机同调的稳定性结果，（2）构造进一步的赋范谱，例如表示Hermitian K-理论的动机谱KO上的赋范谱结构。 另一组部分探讨了从不稳定到稳定的动机同伦理论。尤其是私家侦探。本文将（3）研究动机Barratt-Priddy-Quillen映射，（4）研究一个不稳定的动机同调Whitehead定理。 这些目标将通过使用从更高的范畴理论，经典的稳定同伦理论，代数拓扑和代数几何的技术来实现。第（1）和（2）部分的结果可以用来更有效地研究代数簇使用上同调理论;例如，通过利用上同调运算中存在的理论表示的赋范谱。第（3）和（4）部分对研究motivic（稳定）同伦范畴本身更有用，因此可以同时研究代数簇的所有上同调理论。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.