New Perspectives on Buildings, Geometric Invariant Theory and Algebraic Groups

这个建议涉及到一个叫做群论的数学领域。群是作为一种数学上描述自然界对称性的方式而出现的：这些对称性可以是像晶体结构那样明显的对称性，也可以是不那么明显的对称性，比如描述我们周围世界的方程中固有的对称性。群是19世纪由法国数学家伽罗瓦（Galois）发明的，用来描述何时可以解多项式方程，但不久之后，群理论开始在数学和科学领域得到应用。这个提议中的“代数群”在经典上是作为作用于某个坐标空间的矩阵群而出现的（因此与物理学中使用的群密切相关），但是这些年来已经发展了其他方法来研究这些群。另一位法国数学家雅克·提茨（Jacques Tits）在20世纪50年代末发明了一种叫做“建筑物”的物体，取得了关键进展。这些建筑是高度复杂的对称物体，可以分解成非常简单的碎片（想想DNA这样的复杂分子，由相对简单的构建块组成）；这些部分很容易单独理解，但它们组合在一起的方式产生了一些极其丰富和美丽的数学。它表明，每一个代数群都与其中一个建筑相连，相反，某一类建筑自然也与代数群相连。建筑和群体之间的密切关系使得人们可以将群体理论和相关主题中的问题转化为关于建筑的问题，反之亦然，这一过程是本提案的关键主题之一。自从Tits发明建筑以来，这个提议的核心问题就一直存在，它涉及到建筑物的对称性可能会使建筑物四处移动的方式。也许用一个例子来描述它是最容易的。想象一个球体，它的对称群由所有的旋转和反射组成，这些旋转和反射使它看起来是一样的；例如，我们可以通过球体的中心绕任何轴旋转。如果我们现在给球体上半部分上色，并观察保持这种上色的对称性，那么我们会发现，这种对称性实际上会固定北极和南极。这是Tits猜想的一个特例，该猜想指出，保留某些颜色的建筑物的对称性应该至少有一个固定点。对Tits猜想的解答将是向前迈出的重要一步，而且不仅仅是在纯群论领域。这个猜想将数学的几个重要领域统一在一个保护伞下，并揭示了表面上看起来不相关的结果之间的深刻联系。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。（当你处理复数而不是实数时）；它们可以描述表征理论的某些方面，表征理论是物理、化学和数学的重要工具；它们可以展示给定群体在给定空间中通过对称行为的可能方式。这个提议中的数学有两个主要目的：第一，利用Tits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；其次，利用这些联系和不同的观点来提供一种新的方法来证明Tits的猜想。这两条道路都提供了令人兴奋和创新的新数学的可能性，这些数学将引起各种数学家的兴趣和使用，并通过他们，更广泛的科学家和实践者。

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.