Analyzing algebraic varieties from the point of view of motivic homotopy theory

代数几何是数学最古老的分支之一，其核心是研究多变量多项式方程组。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（例如，数字或抽象代数结构）附加到空间的研究；这些不变量不应该依赖于空间在不撕裂的情况下被拉动或扭曲的方式。当代数变量具有空间结构时，自然会尝试使用代数拓扑工具对其进行分析。然而，当代数变量没有明显的空间结构时，例如，如果它们出现在算术设置中，则需要一种新的方法。本课题的重点是利用Morel-Voevodsky A^1-同伦理论的框架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，但先验地限制了几何结构，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。更具体地说，PI将研究交换一元环上的线性代数问题，例如，射影模的理论和矩阵的分解。这些结构属于代数k理论领域，并通过Morel-Voeovdsky A^1同伦理论与拓扑和算术问题联系起来。其中，PI将研究以下具体（和经典）问题：给定复杂的代数变化，哪些拓扑向量束允许代数结构？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。

Algebraic geometry, one of the oldest branches of mathematics, is at its core 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 (e.g., numbers or abstract algebraic structures) to spaces; these invariants should not depend on the way a space is pulled or twisted without tearing it. When algebraic varieties have a spatial structure, it is natural to try to analyze them using the tools of algebraic topology. However, when algebraic varieties do not have an obvious spatial structure, e.g., if they arise in arithmetic settings, then a new approach is required. The focus of this project is to analyze algebraic invariants of algebraic varieties using the framework of the Morel-Voevodsky A^1-homotopy theory. 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. More specifically, the PI will study problems in linear algebra over commutative unital rings, for example, the theory of projective modules and decompositions of matrices. These structures lie in the domain of algebraic K-theory and are related to topological and arithmetic questions by means of Morel-Voeovdsky A^1-homotopy theory. Among others, the PI will investigate the following concrete (and classical) question: given a complex algebraic variety, which topological vector bundles admit algebraic structures? By their very nature, such problems 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.