This research studies `algebraic groups', mathematical objects which live at the intersection of algebra and geometry. They are simultaneously `groups' and `varieties'. A group is the mathematical abstraction of the idea of symmetry and a variety is the mathematical collection of solutions to a set of equations. Let us start with groups. Consider a square and its symmetries, geometric operations we can do to it that leave it looking like the square we started with. We can rotate it by 0, 90, 180 or 270 degrees. We can also reflect it in 4 straight lines (2 joining corners to corners and 2 joining sides to sides). And these 8 symmetries are all of them. The mathematical abstraction is to see that to understand what is going on we want to consider the symmetries all together, not just on their own. Group theory is about studying these sets of symmetries and the key idea is that when we apply one symmetry and then apply another one, we get back one of our 8 original symmetries. This is the notion of composition. It enjoys special properties. For example, rotating by 0 degrees before or after another symmetry leaves that symmetry unchanged. We have all interacted with this idea from an early age: the integers form a group where composition is just addition of numbers. There, the identity is 0. If we add 0 to any number, it stays the same!
The study of solutions to equations is a vast area of study and has been a focal point since the inception of mathematics. It is the foundation of modern areas like algebraic geometry and number theory. Fermat's Last Theorem is a very famous example of studying sets of solutions of equations and required incredibly deep techniques in algebraic geometry and number theory. Many of us encounter Pythagoras' Theorem and the equation a^2 + b^2 = c^2 at school. There are many integer solutions to this, known as Pythagorean triples, like (3,4,5) and (5,12,13). As with group theory, mathematicians have found that a good way to study these problems is to look at all solutions at once. A variety is a set of solutions to especially nice equations called polynomials.
The study of algebraic groups is therefore highly intradisciplinary. Indeed, their introduction came from generalising Lie groups, which were an analytic invention to study continuous solutions of differential equations. This research will concentrate on the group theoretic side of the story. Algebraic groups have been classified into families and a fundamental open problem is to understand the `simple' ones. These simple groups are the building blocks of all algebraic groups and the word simple obscures the true nature of them. They are incredibly complicated and have a very rich and deep structure. They deserve studying in their own right, let alone due to the applications to other fields of mathematics.
We will study the structure of these simple algebraic groups, where many open problems remain. Our main focus is studying the close relationship with another key area of modern mathematics, representation theory. To study an object, like an algebraic group, we consider how it acts on something more straightforward, in this case a linear space V. Linear spaces are nice, the world we live in is a linear space and as mathematicians we understand them. To study how a group acts on such a linear space, we consider it as a subgroup of the full group of symmetries of the linear space, called GL(V). The crucial part for this research is that GL(V) is itself an algebraic group. And so the structure of algebraic groups and representation theory are intimately related. This was made precise by J.P. Serre when he introduced the concept of complete reducibility. We will tackle open problems about the subgroup structure of algebraic groups, especially certain classes of subgroups related, through this link, to representations that are not completely reducible, meaning they have a more complicated structure built up from the simpler representations.
