New Perspectives on Buildings, Geometric Invariant Theory and Algebraic Groups

点击翻译按钮获取中文摘要

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.