Characterizing algebraic groups via maximal tori

理解对称性，以及它是如何和为什么在自然界中产生的，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被称为“群”，他们的研究被称为“群论”。群的经典例子是旋转或平移(连续群)和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，这些对象捕捉到了连续群(所谓的李群)和离散群的精神，但它们要普遍得多。在接下来的几十年里，代数群论被用来统一处理代数和数论的几个关键领域，包括二次型理论、中心单代数、带对合的代数和一些非结合代数。 该研究计划的中心是理解代数群本身的本质，以及它们在数学的几个领域中的应用。该项目的主要目标是利用代数群的极大环面和分支轨迹刻画它们的重要结果。回想一下，任何代数群都可以被认为是其简单的子对象的“并”，称为极大环面。一个自然的问题立即出现： 假设两个代数群具有相同的极大环面，我们能说什么呢？ 换言之，用“儿童难题”作类比，我们可以把它重新表述为：如果我们破坏给定群G中极大环面之间的所有联系和关系，并取它们不相交的并，人们可以问如何将这些环面粘合在一起以重建G本身，还可以问我们可以用多少种方法来粘合给定环面的一个家庭以构造一个新的群。 刻划具有相同极大环面的绝对几乎单代数群的问题起源于关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中得到了广泛的关注。这部分是由于新发现的与几何问题的联系，这些几何问题涉及等谱和长度可公度的黎曼流形和局部对称空间，但实际上这类问题也与其他领域有关。 我们打算通过研究代数群的分支行为来解决这个问题。我们期望对于定义在有限生成域上的给定群G，只有有限多个群具有与G相同的分支性质。为了得到这一结果，我们将研究扭矩的不同形式的局部-全局原理。回想一下，Torsor是帮助我们根据一些本地数据构建组的工具。在理解扭量的局部-全局行为方面的任何成功都将引导我们解决代数群和几何理论中许多长期存在的猜想。

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.