CAREER: Vector bundles, rational points and homotopy theory

主要研究者将利用Morel-Voevodsky A^1-同伦理论，将经典的代数-阻塞论的思想转换到代数几何的背景下，研究域上交换单位代数上的投射模以及特殊代数簇上有理点的存在性问题。例如，他将与J.Fasel一起，通过更好地理解穿孔仿射空间的A^1-同伦层，来尝试研究秩低于维度的投射模的分裂问题。他将与C.Haesmeyer一起，再次使用A^1-同伦理论，试图给出域上光滑代数簇上有理点存在的可计算的新障碍。根据它们的性质，这些问题将数学的几个分支结合在一起，从而说明了这门学科的基本统一性。为了使新一代研究人员能够接触到上述研究及相关领域中所使用的技术，PI将组织一系列暑期班，致力于向研究生介绍A^1-同伦理论及相关学科。代数几何是数学中最古老的分支之一，它的核心是研究代数簇，即多变量多项式方程组的解。代数拓扑学研究不变量，例如数字，或者更广泛地说，抽象的代数结构，以一种独立于它们被拉和扭曲的方式(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用于代数拓扑学的工具，特别是当这种方程组是从算术中产生的时候。然而，最近引入的A^1-同伦理论提供了一个框架，在该框架中，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-人们可以处理具有深算术结构但先验有限几何结构的空间，与更本质上几何的空间处于同等地位。这个项目的目的是通过将一种非常成功的方法从代数拓扑(障碍论)移植到代数几何的设置上，借助于A^1-同伦理论来研究两类一般问题。为了让更广泛的观众了解上述研究中涉及的一些想法，国际数学奖将与艺术家蔡伦义合作，在艺术家和数学家之间组织公开的“对话”，以探索主题之间的一些共同主题。该奖项由代数与数论和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.