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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677091
• 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.***********

理解对称性，以及它是如何和为什么在自然界中产生的，对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”。群的经典例子是旋转或平移的群（连续群）和正方形或雪花的对称（离散群）。20世纪中期见证了代数群（Algebraic Groups）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，但它们更加普遍。在随后的几十年里，代数群理论被用于对代数和数论的几个关键领域进行统一的处理，包括二次型理论、中心简单代数、对合代数和一些非结合代数。******该研究项目的中心是理解代数群本身的本质，以及它们在几个数学领域的应用。该项目的主要目标是在通过代数群的最大环面和分支轨迹来表征代数群方面取得重要成果。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”，称为极大环面。一个自然的问题立刻出现了：******对于两个代数群，如果它们有相同的最大环面，我们能说些什么呢？******换句话说，利用“儿童拼图”的类比，我们可以这样表述：如果我们破坏给定群G中最大环面之间的所有连接和关系，并取它们的不相交并，我们可以问如何将这些环面粘合在一起以重建G本身。同样，我们可以问有多少种方法可以粘合一个给定的tori家族来构建一个新的群体。******具有相同极大环面的绝对几乎简单代数群的刻画问题源于除法代数的极大子域和分裂域的经典结果，近年来在代数和几何领域受到了广泛的关注。这部分是由于新发现的与几何问题的联系，涉及等谱和长度可通约黎曼流形和局部对称空间，但事实上，这类问题也与其他领域相关。******我们打算通过研究代数群的分枝行为来解决这个问题。我们期望，对于在有限生成场上定义的给定群G，只有有限多个群具有与G相同的分支性质。为了得到这个结果，我们将研究不同形式的局部-全局原理。回想一下，tor是帮助我们从一些本地数据构建组的工具。任何对局部-全局行为的理解的成功都将引导我们解决代数群和几何理论中许多开放的长期猜想。***********