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。这样的群以各种物体的对称群的形式出现，并且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。在算术方面，过去六十年的工作已经导致了有理数和其他类似领域的线性代数群理论的发展。虽然这一领域的活动仍在进行中，但在过去十年中，李群论、算术几何和其他学科中的各种问题引起了人们对几何原点域上代数群的性质的极大兴趣。在先前工作的基础上，该研究项目将研究这些高维域上代数群的算术、几何和结构方面，特别关注各种有限性质。指导研究生和开发本科和研究生水平的课程将是这项工作的组成部分。此外，还将开展一项图书计划，向更广泛的读者介绍高维领域代数群的新兴算术理论的最新发展。该项目是研究高维域上代数群的一个多方面的研究项目。研究工作将集中在以下三个方向：具有良好约简性的代数群的分析和局部-全局原理的应用，非分枝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。具有良好约简的群的研究的一个主要目标将是在有限生成域上具有良好约简的约简代数群的形式的有限猜想上取得进展。这项工作将极大地扩展先前主要处理Dedekind环分数域上群的结果的范围，并将对代数群的伽罗瓦上同调中全局到局部映射的正确性产生重要影响。结果表明，对于某些类型的群，这个有限性猜想与未分叉上同调的有限性密切相关。因此，其中一个目标将是在全球范围内建立曲面和某些高维品种的三度非分支上同调的预期有限。关于抽象同态，目标将是对以前工作中介绍的方法进行实质性的推广，以解决一个长期存在的Borel和Tits猜想，该猜想适用于所有相对秩至少为2的无限域上的绝对几乎单群。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.