New Perspectives on Buildings, Geometric Invariant Theory and Algebraic Groups

建筑、几何不变量理论和代数群的新视角

• 批准号: EP/L005328/1
• 负责人: Michael Bate
• 金额: $ 38万
• 依托单位: University of York
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL005328%2F1
• 关键词: New; Perspectives; Buildings; Geometric; Invariant

This proposal concerns an area of mathematics called group theory. Groups arise as a way of mathematically describing symmetries which occur in nature: these could be obvious symmetries like those in crystal structures, or less obvious symmetries such as those inherent in equations describing the world around us. Groups were invented in the 1800s by a French mathematician called Galois as a way of describing when it is possible to solve polynomial equations, but before long the theory of groups started finding applications across mathematics and science. The "algebraic groups" in this proposal arise classically as groups of matrices acting on some space of coordinates (and are hence strongly related to groups used in physics), but other approaches to these groups have developed over the years. A key advance was made by another French mathematician, Jacques Tits, when he invented objects called "buildings" in the late 1950s. These buildings are highly complicated symmetric objects which break up into very simple pieces (think of a complex molecule like DNA, made up of relatively simple building blocks); the pieces are easy to understand individually, but the way they fit together gives rise to some extremely rich and beautiful mathematics. Tits showed that every algebraic group has attached to it one of these buildings and, conversely, a certain class of buildings naturally have attached to them algebraic groups. The close relationship between buildings and groups allows one to translate problems in group theory and related topics into to problems about buildings, and vice versa, and this process is one of the key themes of this proposal. The question at the heart of this proposal has been around since Tits invented buildings and concerns the possible ways that symmetries of a building can move the building around. It is perhaps easiest to describe with an example. Imagine a sphere, whose group of symmetries consists of all rotations and reflections which leave it looking the same; for example, we're allowed to rotate the sphere about any axis through its centre. If we now colour the top half of the sphere and just look at the symmetries which preserve this colouring, then we see that such symmetries will in fact fix the north and south poles. This is a special case of Tits' conjecture, which states that the symmetries of a building preserving certain colourings should have at least one fixed point. A solution to Tits' conjecture would be a major step forward, and not just in pure group theory. The conjecture unifies several important areas of mathematics under one umbrella, and exposes deep connections between results which on the surface appear unrelated. For example, buildings can be used to encode what happens when you change your number system (eg., when you work with complex numbers instead of real numbers); they can describe some aspects of representation theory, which is a vital tool in physics and chemistry as well as mathematics; they can exhibit the possible ways that a given group can act by symmetries on a given space. The mathematics in this proposal has two main aims: first, to use the context provided by Tits' conjecture to develop new connections and results within group theory and other related areas; second, to exploit these connections and different points of view to give a novel approach to proving Tits' conjecture. Both these paths offer the possibility of exciting and innovative new mathematics which will be of interest and use to a wide variety of mathematicians and, through them, a wider audience of scientists and practitioners.

这个建议涉及一个叫做群论的数学领域。群是作为一种数学描述自然界中出现的对称性的方式而出现的：这些对称性可以是像晶体结构中的对称性那样明显的对称性，也可以是不太明显的对称性，比如描述我们周围世界的方程中所固有的对称性。群是在19世纪由法国数学家伽罗瓦（Galois）发明的，作为描述何时可以求解多项式方程的一种方法，但不久之后，群的理论开始在数学和科学中得到应用。这个提议中的“代数群”经典地作为作用于某些坐标空间的矩阵群出现（因此与物理学中使用的群密切相关），但这些群的其他方法已经发展多年。另一位法国数学家雅克·蒂茨（Jacques Tits）在20世纪50年代末发明了一种称为“建筑物”的物体，取得了关键性的进展。这些建筑物是高度复杂的对称物体，它们被分解成非常简单的碎片（想想像DNA这样的复杂分子，由相对简单的积木组成）;这些碎片很容易单独理解，但是它们组合在一起的方式产生了一些非常丰富和美丽的数学。山雀表明，每一个代数组已重视它的一个这些建筑物，反过来说，一定类的建筑物自然有重视他们的代数组。建筑物和群体之间的密切关系使人们能够将群论和相关主题中的问题转化为关于建筑物的问题，反之亦然，这个过程是本提案的关键主题之一。这一提议的核心问题自山雀发明建筑物以来就一直存在，并涉及建筑物的对称性可以移动建筑物的可能方式。用一个例子来描述可能是最容易的。想象一个球体，它的对称群由所有旋转和反射组成，这些旋转和反射使它看起来相同;例如，我们可以围绕通过其中心的任何轴旋转球体。如果我们现在给球的上半部分着色，并只观察保持这种着色的对称性，那么我们就会看到，这种对称性实际上会固定北极和南极。这是山雀猜想的一个特例，山雀猜想指出，一座建筑物的对称性保留了某些颜色，应该至少有一个不动点。解决山雀猜想将是一个重大的进步，而不仅仅是在纯粹的群论。该猜想将数学的几个重要领域统一在一个保护伞下，并揭示了表面上看似无关的结果之间的深层联系。例如，建筑物可以用来编码当你改变你的数字系统时发生的事情（例如，当你处理复数而不是真实的数时）;它们可以描述表示论的某些方面，这是物理和化学以及数学中的重要工具;它们可以展示给定群在给定空间上通过对称性起作用的可能方式。这个提议中的数学有两个主要目的：第一，利用山雀猜想提供的背景，在群论和其他相关领域中发展新的联系和结果;第二，利用这些联系和不同的观点，给出一种新的方法来证明山雀猜想。这两条道路都提供了令人兴奋和创新的新数学的可能性，这将是各种各样的数学家感兴趣和使用，并通过他们，更广泛的科学家和从业者的观众。

项目成果

Orbit closures and invariants

轨道闭合和不变量

• DOI: 10.1007/s00209-019-02228-6
• 发表时间: 2019
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Bate M

$G$-complete reducibility in non-connected groups

$G$-非连接组中的完全可还原性

• DOI: 10.1090/s0002-9939-2014-12348-3
• 发表时间: 2014
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Bate M

On a question of Külshammer for representations of finite groups in reductive groups

关于还原群中有限群表示的 Külshammer 问题

• DOI: 10.1007/s11856-016-1337-2
• 发表时间: 2016
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: Bate M

Cocharacter-closure and the rational Hilbert-Mumford Theorem

共字符闭合和有理 Hilbert-Mumford 定理

• DOI: 10.1007/s00209-016-1816-5
• 发表时间: 2016
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Bate M

Composition Factors of Tensor Products of Symmetric Powers

对称幂张量积的构成因子

• DOI: 10.48550/arxiv.1704.02410
• 发表时间: 2017
• 作者: Donkin S