Classical Methods in Motivic Homotopy Theory

代数几何是研究多项式方程组的学科。事实证明，除了少数非常特殊的情况外，准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。一种经典的卓有成效的方法是将解集合视为一个拓扑空间，一个位于某个高维空间中的对象。例如，方程x^2 y^2=1描述了一个半径为1的圆。拓扑学家已经发明了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不变量。把这个想法发挥到极致，我们就会进入一个叫做理据同伦理论的领域。它提供了定性描述多项式方程组解的新方法，并在过去的二十年中成功地应用于代数几何中的几个长期开放的问题。这个项目旨在通过进一步探索它与经典拓扑学的相似之处，以及这些相似之处发生了什么，来加深我们对动机同伦理论的理解。本项目由几个部分组成。其中一组部分探索了Motivic高结构环谱的性质，称为范数谱(与M.Hoyois合作介绍)。首席调查员(P.I.)将(1)研究赋范谱中的幂运算，并推导正特征赋范谱的分裂结果和对称群的基元同调的稳定性结果，以及(2)构造进一步的赋范谱，例如表示厄米特K-理论的基元谱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。具体地说，P.I.将(3)研究Motivic Barratt-Priddy-Quillen映射，(4)研究一个不稳定的Motivic同调Whitehead定理。这些目标将通过使用更高范畴理论、经典稳定同伦理论、代数拓扑学和代数几何的技巧来实现。第(1)和(2)部分中的结果可用于使用上同调理论更有效地研究代数簇；例如，通过利用赋范谱表示的理论中存在的上同调运算。第(3)和(4)部分更有助于研究基元(稳定)同伦范畴本身，从而一次性总结所有代数变种的上同调理论。这一奖项反映了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。

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.