Geometric Applications of Motivic Homotopy Theory

该奖项支持代数几何的研究，这是一个与多变量多项式方程系统研究有关的主题。 这些系统的解产生了代数簇，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（数字或抽象代数结构）附加到空间的研究;这些不变量不依赖于空间被拉伸或扭曲的方式（不撕裂它）。 试图用代数拓扑学的工具来分析代数簇是很自然的。 然而，当代数簇没有明显的“空间结构”时，例如当它们出现在算术设置中时，则需要新的方法。 正在考虑的项目旨在使用一个新的框架来分析代数簇的代数不变量。 这一理论允许一个应用的全部权力的技术代数拓扑的对象感兴趣的代数几何-一个可以治疗空间具有复杂的算术结构，但先验有限的几何结构，在本质上相同的方式作为更经典的空间。 本项目旨在利用这些新技术分析某些经典的代数和算术问题，并提供更好的理解代数方程的具体系统，这是数学的许多领域的基础。研究者将研究关于投射模的问题交换酉代数域上，以及相关的拓扑和算术问题，通过Morel-Voevodsky A1-同伦理论。 在其他主题中，该项目调查了以下具体（和经典）问题。 给定一个复代数簇，哪些拓扑向量丛具有代数结构？ 有限域上的光滑仿射簇上有多少个代数向量丛（假设有固定不变量）？ 就其本质而言，这些问题汇集了数学的几个分支，从而说明了该学科的基本统一性。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.