Rationality problems and homotopy theory for varieties

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首席研究员将研究代数几何中经典研究对象的算术性质问题，即所谓的有理数和近有理数的变化，通过基于拓扑的极其精细的不变量来研究它们的几何。他将与F. Morel一起，利用最近引进的受著名的Browder-Novikov-Sullivan-Wall流形分类启发的技术，研究对这些变体进行分类的问题。与B. Doran一起，他将继续研究单能群作用的不变理论，它与A^1可收缩变的构造的关系，以及将仿射空间刻画为代数变的问题。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。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。代数拓扑研究附加不变量的问题，例如，数字，或者更一般地，所谓的抽象代数结构到空间；这些不变量最重要的性质是，只要在这个过程中没有被撕裂，它们就不会随着底层几何对象被拉动和扭曲而改变。代数几何中的一个基本问题是代数变量的分类，即解的可能组态的显式确定和分类。当方程组的解具有拓扑结构时（想想定义球体的方程），人们可以尝试用不变量来区分它们。然而，在算术中出现的多项式方程组的解并不总是具有明显的拓扑结构（想想定义单位球的方程的整数解）。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变体——从不变量的角度来看，具有深度算术（但先验的有限几何）结构的空间与那些本质上更几何的空间处于同等地位。这个项目的目的是通过移植一种非常成功的方法，通过外科手术对拓扑空间进行分类，这是一种剪切和粘贴过程，非常仔细地由适当的不变量控制，通过a ^1同伦理论将算法、拓扑和代数几何的思想进一步融合到代数几何中。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The Principal Investigator will work on problems relating arithmetic properties of classical objects of study in algebraic geometry, i.e., so-called rational and nearly rational varieties, to their geometry by means of extremely refined invariants rooted in topology. Together with F. Morel, he will investigate the problem of classifying such varieties using recently introduced techniques inspired by the celebrated Browder-Novikov-Sullivan-Wall classification of manifolds. Together with B. Doran, he will continue to investigate invariant theory for unipotent group actions, its relationship to construction of A^1-contractible varieties, and the problem of characterizing affine space as an algebraic variety. By their very nature, these problems draw together several branches of mathematics and thus illustrate the fundamental unity of the subject. 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, so-called abstract algebraic structures to spaces; the most important property of such invariants is that they do not change as the underlying geometric object is pulled and twisted, so long as it is not torn in the process. A fundamental problem in algebraic geometry is classification of algebraic varieties, i.e., explicit determination and taxonomy of the possible configurations of solutions. When solutions to systems of equations have topological structure (think of the equation defining a sphere), one can try to distinguish them by means of invariants. However, solutions to systems of polynomial equations arising in arithmetic do not always have obvious topological structure (think of the integer solutions to the equation defining a unit sphere). 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 algebraic varieties--spaces having deep arithmetic (but a priori limited geometric) structure are, from the standpoint of invariants, put on equal footing with those that are more inherently geometric. The aim of this project is to further amalgamate ideas of arithmetic, topology, and algebraic geometry by transplanting the fantastically successful method of classifying topological spaces via surgery, a cutting and pasting procedure very carefully controlled by appropriate invariants, into algebraic geometry by means of A^1-homotopy theory.