CAREER: Vector bundles, rational points and homotopy theory

利用Morel-Voevodsky的a ^1同伦理论，将经典代数拓扑障碍理论的思想转到代数几何的背景下，研究域上交换一元代数上的射射模问题，以及特殊代数变异类上有理点的存在性问题。例如，他将与J. Fasel一起，通过更好地理解刺穿仿射空间的A^1同伦束，尝试研究维数以下秩的投影模的分裂问题。他将与C. Haesemeyer一起，再次利用a ^1同伦理论，试图在域上光滑代数变异上产生可计算的新的有理点存在障碍。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。为了使上述研究和相关领域中使用的技术能够被新一代的研究人员使用，PI将组织一系列的暑期学校，致力于向研究生介绍a ^1同伦理论和相关学科。代数几何是数学最古老的分支之一，其核心是对代数变量的研究，即多变量多项式方程组的解。代数拓扑学研究将不变量，例如数字或更一般的抽象代数结构，以独立于它们被拉和扭曲（不被撕裂）的方式附加到空间的问题。多项式方程组的解显然不适合于代数拓扑工具，特别是当这类方程组由算术产生时。然而，最近引入的A^1同伦理论提供了一个框架，在这个框架中，人们可以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——人们可以将具有深度算术结构的空间，但先验的有限几何结构，与更固有的几何空间平等地对待。本课题利用a ^1同伦理论，将一种非常成功的代数拓扑方法（障碍物理论）移植到代数几何的集合中，研究两类一般问题。为了让更多的人了解上述问题，PI将与艺术家蔡伦毅合作，组织艺术家与数学家之间的公开“对话”，探讨两者之间的共同主题。该奖项由代数和数论以及拓扑学项目共同资助。

The Principal Investigator will study problems regarding projective modules over commutative unital algebras over a field, and existence of rational points on special classes of algebraic varieties by transposing ideas from the classical algebro-topological theory of obstructions to the context of algebraic geometry by means of the Morel-Voevodsky A^1-homotopy theory. Together with J. Fasel, he will, for example, attempt to study splitting problems for projective modules of rank below the dimension by better understanding A^1-homotopy sheaves of punctured affine spaces. Together with C. Haesemeyer, he will attempt to produce computable new obstructions to existence of a rational point on smooth algebraic varieties over a field, again using A^1-homotopy theory. By their very nature, these problems draw together several branches of mathematics and thus illustrate the fundamental unity of the subject. To make the techniques used in the above study and related areas accessible to a new generation of researchers, the PI will organize a series of summer schools devoted to introducing graduate students to A^1-homotopy theory and related subjects. Algebraic geometry, one of the oldest branches of mathematics, is at its core concerned with the study of algebraic varieties, i.e., solutions to systems of polynomial equations in many variables. Algebraic topology studies the problem of attaching invariants, e.g., numbers or, more generally, abstract algebraic structures, to spaces in a manner independent of the way they are pulled and twisted (without being torn). Solutions to systems of polynomial equations are not obviously amenable to the tools of algebraic topology, especially when such systems arise from arithmetic. Nevertheless, the relatively recently introduced subject of A^1-homotopy theory provides a framework in which one may apply the full power of techniques of algebraic topology to objects of interest in algebraic geometry---one can treat spaces having deep arithmetic structure, but a priori limited geometric structure, on equal footing with spaces that are more inherently geometric. This project aims to study two general classes of problems by transplanting a very successful method from algebraic topology (the theory of obstructions) to the setting of algebraic geometry by means of A^1-homotopy theory. With the goal of making some ideas involved in the above pursuits accessible to a broader audience, the PI will collaborate with artist Lun-Yi Tsai to organize public ``dialogues" between artists and mathematicians to explore some common themes between the subjects.This award is cofunded by the Algebra and Number Theory and the Topology programs.