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。主要研究者将致力于研究代数几何中经典研究对象的算术性质问题，即，所谓的理性和近理性的品种，他们的几何通过非常精致的不变量植根于拓扑。 连同楼莫雷尔，他将调查的问题，分类等品种使用最近推出的技术启发著名的Browder-Novikov-Sullivan-Wall分类的流形。 和B一起。多兰，他将继续调查不变量理论的unipotent组行动，其关系建设的A^1-收缩品种，以及问题的特点仿射空间作为一个代数品种。 就其本质而言，这些问题将数学的几个分支结合在一起，从而说明了该学科的基本统一性。 代数几何是数学最古老的分支之一，其核心是研究代数簇，即，多变量多项式方程组的解。 代数拓扑学研究附加不变量的问题，例如，这种不变量的最重要的性质是，只要在这个过程中不被撕裂，它们就不会随着底层几何对象的拉伸和扭曲而改变。 代数几何中的一个基本问题是代数簇的分类，即，明确确定和分类的可能配置的解决方案。 当方程组的解具有拓扑结构时（考虑定义球面的方程），人们可以尝试通过不变量来区分它们。 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（想想定义单位球面的方程的整数解）。 尽管如此，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簇--从不变量的观点来看，具有深度算术（但先验有限几何）结构的空间与那些更内在几何的空间处于平等的地位。 这个项目的目的是通过A^1-同伦理论将非常成功的拓扑空间分类方法移植到代数几何中，进一步融合算术、拓扑和代数几何的思想。

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.