Arithmetic Questions in the Theory of Linear Algebraic Groups

线性代数群是由多项式方程描述的矩阵群。这种群以各种对象的对称性群的形式出现，在数学的许多领域中无处不在，包括代数几何、数论和数学物理。在算术背景下，过去60年的工作已经导致了关于有理数和其他类似领域上的线性代数群的良好发展的理论。虽然这一领域的活动仍在进行中，但在过去的十年里，李群理论、算术几何和其他学科的各种问题已经引起了人们对几何起源域上代数群的性质的极大兴趣。在以前工作的基础上，该研究计划将研究此类高维域上代数群的算术、几何和结构方面，并特别关注各种有限性质。指导研究生和开发本科生和研究生课程将是这项工作不可或缺的一部分。此外，还将进行一项图书项目，向更广泛的读者介绍高维域上代数群的新兴算术理论的最新发展。该项目是一个多方面的研究计划，研究高维域上的代数群。这项工作将集中在以下三个方向：具有良好归约和应用于局部-整体原理的代数群的分析，未分支上同调的有限性质的研究，以及代数群抽象同态的刚性现象的研究。关于具有良好约化的群的研究的一个主要目标将是在关于关于离散赋值的除子集具有良好约化的有限生成域上的约化代数群的形式的有限猜想上取得进展。这项工作将大大扩展以前主要研究Dedekind环的分数域上的群的结果的范围，并且对代数群的Galois上同调中的全局到局部映射的适定性也有重要的结果。结果表明，对于某些类型的群，这个有限猜想与未分支上同调的有限性质密切相关。因此，目标之一将是为全球场上的曲面和某些高维变体建立预期的三度未分支上同调的有限性。关于抽象同态，目标将是对以前工作中引入的方法进行实质性的推广，以解决Borel和Tits的一个长期猜想，该猜想适用于至少两个相对等级的无限域上的所有绝对简单群。该奖项反映了NSF的法定使命，并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，被认为值得支持。

Linear algebraic groups are groups of matrices that are described by polynomial equations. Such groups arise as groups of symmetries of various objects and are ubiquitous across many areas of mathematics, including algebraic geometry, number theory, and mathematical physics. In the arithmetic context, work done over the last six decades has resulted in a well-developed theory of linear algebraic groups over the rational numbers and other similar fields. While activity in this area is still ongoing, over the last ten years various problems in Lie group theory, arithmetic geometry, and other subjects have led to significant interest in the properties of algebraic groups over fields of geometric origin. Building on previous work, the research program will investigate the arithmetic, geometric, and structural aspects of algebraic groups over such higher-dimensional fields, with a particular focus on various finiteness properties. Mentoring graduate students and developing courses at the undergraduate and graduate levels will be an integral part of this work. In addition, a book project will be undertaken to open up recent developments in the emerging arithmetic theory of algebraic groups over higher-dimensional fields to a broader audience. The project is a multi-faceted research program in the study of algebraic groups over higher-dimensional fields. The work will focus on the following three directions: the analysis of algebraic groups with good reduction and applications to local-global principles, the study of finiteness properties of unramified cohomology, and the investigation of rigidity phenomena for abstract homomorphisms of algebraic groups. A major goal in the study of groups with good reduction will be to make progress on a finiteness conjecture for forms of reductive algebraic groups over finitely generated fields having good reduction with respect to divisorial sets of discrete valuations. This work will significantly expand the scope of previous results, which dealt mainly with groups over fraction fields of Dedekind rings, and will also have important consequences for the properness of the global-to-local map in the Galois cohomology of algebraic groups. It turns out that, for certain types of groups, this finiteness conjecture is closely related to finiteness properties of unramified cohomology. As a result, one of the objectives will be to establish the expected finiteness of unramified cohomology in degree three for surfaces and certain higher-dimensional varieties over global fields. Concerning abstract homomorphisms, the goal will be to develop a substantial generalization of methods introduced in previous work to resolve a longstanding conjecture of Borel and Tits for all absolutely almost simple groups over infinite fields of relative rank at least two.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.