Geometric Applications of Motivic Homotopy Theory

该奖项支持对代数几何的研究，这是一门研究多变量多项式方程组的学科。这些方程组的解产生代数变分，代数变分是代数几何的基本研究对象。代数拓扑学是系统地将代数不变量（数字或抽象代数结构）附加到空间上的研究；这样的不变量不依赖于空间被拉动或扭曲的方式（不撕裂它）。利用代数拓扑的工具来分析代数变量是很自然的。然而，当代数变量没有明显的“空间结构”时，例如当它们出现在算术设置中时，则需要一种新的方法。正在考虑的项目寻求使用一个新的框架来分析代数变量的代数不变量。该理论允许人们将代数拓扑技术的全部力量应用于代数几何中感兴趣的对象-人们可以处理具有复杂算术结构的空间，但先验的几何结构有限，本质上与更经典的空间相同。目前的项目旨在利用这些新技术分析某些经典代数和算术问题，并提供对代数方程的特定系统的更好理解，这是许多数学领域的基础。研究者将利用Morel-Voevodsky a1 -同伦理论研究域上交换一元代数上的射影模问题以及相关的拓扑和算术问题。在其他主题中，该项目调查了以下具体（和经典）问题。给定一个复杂的代数变化，哪些拓扑向量束承认代数结构？在有限域上的光滑仿射变化（比如固定不变量）上有多少代数向量束？就其本质而言，这些问题汇集了数学的几个分支，从而说明了这门学科的基本统一性。该奖项反映了美国国家科学基金会的法定使命，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。

This award supports research into algebraic geometry, a subject concerned with the study of systems of polynomial equations in many variables. The solutions to such systems give rise to algebraic varieties, which are fundamental objects of study in algebraic geometry. Algebraic topology is the study of systematically attaching algebraic invariants (numbers or abstract algebraic structures) to spaces; such invariants do not depend on the way a space is pulled or twisted (without tearing it). It is natural to try to analyze algebraic varieties using the tools of algebraic topology. However, when algebraic varieties do not have obvious "spatial structure," such as when they arise in arithmetic settings, then a new approach is required. The project under consideration seeks to analyze algebraic invariants of algebraic varieties using a new framework. This theory allows one to apply the full power of techniques of algebraic topology to objects of interest in algebraic geometry -- one may treat spaces having complicated arithmetic structure, but a priori limited geometric structure, in essentially the same way as more classical spaces. The current project seeks to analyze certain classical algebraic and arithmetic questions using these new techniques and to provide a better understanding of specific systems of algebraic equations, which is fundamental to many areas of mathematics.The investigator will study questions regarding projective modules over commutative unital algebras over a field, and related topological and arithmetic questions, by means of the Morel-Voevodsky A1-homotopy theory. Among other topics, the project investigates the following concrete (and classical) questions. Given a complex algebraic variety, which topological vector bundles admit algebraic structures? How many algebraic vector bundles are there on a smooth affine variety over a finite field (say with fixed invariants)? By their very nature, these questions draw together several branches of mathematics and thus illustrate the fundamental unity of the subject.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.