Characterizing algebraic groups via maximal tori

通过最大环面表征代数群

• 批准号: RGPIN-2017-05749
• 负责人: Chernousov, Vladimir
• 金额: $ 1.46万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711784
• 关键词: Characterizing; algebraic; groups; via; maximal

Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. Recall that the mathematical objects that measure symmetry are called ``groups'' and their study is known as ``group theory''. Classical examples of groups are those of rotations or translations (continuous groups) and the symmetry of a square or a snowflake (discrete groups). The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of continuous groups (the so-called Lie groups) and discrete groups, but that are much more universal. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key areas of algebra and number theory, including the theories of quadratic forms, central simple algebras, algebras with involution and some non-associative algebras. The research program centers on understanding the very nature of algebraic groups themselves, and their applications to several areas of Mathematics. The main goal of the project is to achieve important results in characterizing algebraic groups via their maximal tori and ramification locus. Recall that any algebraic group can be thought of as a ``union of its simple subobjects", called maximal tori. A natural question appears immediately: What can one say about two algebraic groups given that they have the same maximal tori? In other words, using analogy with ``children puzzles", we can rephrase it as follows: if we destroy all connections and relations between maximal tori in a given group G and take their disjoint union, one can ask how to glue these tori together in order to reconstruct G itself. Also, one can ask in how many ways we can glue a family of given tori in order to construct a new group. The problem of characterizing absolutely almost simple algebraic groups having the same maximal tori is rooted in the classical results on the maximal subfields and the splitting fields of division algebras and it has recently received a good deal of attention in algebra and geometry. This was due in part to newly discovered connections with geometric problems involving isospectral and length-commensurable Riemannian manifolds and locally symmetric spaces, but in fact questions of this kind are relevant also for other areas. We intend to attack this problem by the studying the ramification behavior of algebraic groups. We expect that for a given group G defined over a finitely generated field there are only finitely many groups which have the same ramification properties as G . To obtain this result we are going to study different forms of local-global principles for torsors. Recall that torsors are tools that help us to construct groups out of some local data. Any success in understanding local-global behavior of torsors would lead us to solutions of many open long-standing conjectures in the theory of algebraic groups and geometry.

了解对称性，以及它在自然界中如何以及为何出现，对于数学和物理学都很重要。回想一下，测量对称性的数学对象被称为“群”，它们的研究被称为“群论”。群的经典示例是旋转或平移（连续群）以及正方形或雪花的对称性（离散群）。 20 世纪中叶见证了代数群的诞生，这些对象捕捉了连续群（所谓的李群）和离散群的精神，但更加普遍。在随后的几十年中，代数群理论被用来统一处理代数和数论的几个关键领域，包括二次形式、中心简单代数、对合代数和一些非结合代数的理论。 该研究项目的重点是了解代数群本身的本质及其在数学多个领域的应用。该项目的主要目标是通过代数群的最大环面和分枝轨迹来表征代数群，取得重要成果。回想一下，任何代数群都可以被认为是“其简单子对象的并集”，称为最大环面。一个自然的问题立即出现： 考虑到两个代数群具有相同的最大环面，我们能说些什么呢？ 换句话说，使用“儿童拼图”进行类比，我们可以将其改写如下：如果我们破坏给定群 G 中最大环面之间的所有联系和关系，并取它们的不相交并集，人们可以问如何将这些环面粘合在一起为了重建 G 本身，我们还可以问我们可以用多少种方法来粘合一组给定的环面以构造一个新的群。 表征具有相同最大环面的绝对几乎简单代数群的问题植根于除法代数的最大子域和分裂域的经典结果，并且最近在代数和几何中受到了广泛的关注。这部分是由于新发现的与涉及等谱和长度可通黎曼流形以及局部对称空间的几何问题的联系，但实际上此类问题也与其他领域相关。 我们打算通过研究代数群的分支行为来解决这个问题。我们期望对于在有限生成域上定义的给定群 G ，只有有限多个具有与 G 相同的分支属性的群。为了获得这个结果，我们将研究不同形式的 Torsors 局部全局原则。回想一下，tors 是帮助我们根据一些本地数据构建组的工具。任何成功地理解扭转量的局部-全局行为都将引导我们解决代数群和几何理论中许多长期存在的开放猜想。