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

代数几何是数学中最古老的分支之一，它的核心是研究多变量多项式方程组。这类系统的解产生了代数变分，这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，数字或抽象的代数结构)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或扭曲的方式。当代数簇具有空间结构时，尝试用代数拓扑学的工具来分析它们是很自然的。然而，当代数变体不具有明显的空间结构时，例如，如果它们出现在算术设置中，则需要新的方法。这个项目的重点是利用Morel-Voevodsky A^1-同伦理论的框架来分析代数簇的代数不变量。这一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但先验有限几何结构的空间，其方式基本上与更经典的空间相同。目前的项目试图使用这些新技术分析某些经典的代数和算术问题，并更好地理解特定的代数方程系统，这是许多数学领域的基础。更具体地说，PI将研究交换单位环上的线性代数中的问题，例如投射模理论和矩阵分解。这些结构属于代数K-理论的范畴，并且借助于Morel-Voeovdsky A^1-同伦理论与拓扑和算术问题有关。其中，PI将研究以下具体的(和经典的)问题：给定一个复杂的代数簇，哪些拓扑向量丛具有代数结构？就其本质而言，这类问题将数学的几个分支结合在一起，从而说明了学科的基本统一性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.