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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=647480
• 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，以及它在自然界中的出现以及为什么在数学和物理学中都很重要。回想一下，测量对称性的数学对象称为``组''，他们的研究被称为````组理论'''。组的经典示例是旋转或翻译（连续组）的示例和正方形或雪花的对称性（离散组）。 20世纪中叶，代数群体的诞生，捕捉连续群体（所谓的谎言群体）的精神的物体和离散的群体的诞生，但更普遍。在随后几十年的过程中，代数群体的理论已被用来对代数的几个关键领域和数字理论进行统一处理，包括二次形式的理论，中央简单代数，代数，具有相关性和一些非缔合性代数。 *****研究计划以理解代数群体本身的本质以及它们在数学几个领域的应用中为中心。该项目的主要目标是通过其最大托里和后果位点来表征代数群体来取得重要的结果。回想一下，任何代数群体都可以被认为是``简单的亚物体的结合''，称为最大托里。换句话说，最大的托里呢？人们可以问如何将这些摩托车粘合在一起，以重建g本身。另外，人们可以通过以多种方式询问一个给定托里家族以构建一个新群体的方式。 ******表征具有相同最大托里的绝对几乎简单的代数组的问题植根于最大子字段和分区代数的分裂场的经典结果，并且最近在代数和代数和代数中受到了广泛关注。几何学。这部分是由于新发现的与涉及同一和长度可承受的riemannian流形和本地对称空间的几何问题的联系，但实际上，这种问题也与其他领域有关。 ******我们打算通过研究代数群体的后果行为来攻击这个问题。我们希望，对于在有限生成的字段上定义的给定G的G，只有许多有限的组具有与G相同的分支属性。为了获得此结果，我们将研究不同形式的Torsors局部全球原则。回想一下，Torsors是帮助我们从某些本地数据中构建组的工具。理解扭矩的局部全球行为的任何成功都将使我们在代数群体和几何学理论中为许多开放的长期猜想提供解决方案。***************************************************