### Clinton Boys

Australian software engineer and mathematician, living in London.

This post is an attempt for me to organise in my own head an answer to a question I’m asked all the time: what was your PhD about?

What did you study?

I studied pure mathematics. Research mathematics is traditionally divided into two areas: applied mathematics, which deals with equations and concepts motivated from other sciences and from modelling real-world phenomena, and pure mathematics, which deals with studying mathematics for its own sake. Sometimes the two branches cross over, and sometimes mathematics that was once considered pure is later found to have applications.

My branch of pure mathematics was algebra. Algebra is the study of mathematical structure. When algebraists talk about structure, they mean the kind of structure that numbers and number systems have. For example, the integers, with their operations of addition and negation, form a type of algebraic structure known as a group. There are many other types of groups, and algebraists (specifically group theorists) busy themselves with the task of classifying groups: that is, coming up with a list of all possible group structures. Classification is a general mathematical principle that says it is interesting to completely enumerate or describe all possible structures arising from a certain series of assumptions. The classification of group structures is a monumentally difficult task that was only (partially) solved in the last decade.

Certain groups are particularly interesting in their own right. One of these groups is called the symmetric group. It has been studied for over a century and can be found in physics, chemistry and other places in nature. The symmetric group has given rise to its own branch of algebra known as representation theory: this was the sub-branch of algebra that I studied.

Specifically, I was concerned with the classification of representations of a family of algebraic objects related to symmetric groups. Two American professors came up with a groundbreaking result about six years ago showing that you could understand these objects in a different way, because they were actually the same as another collection of objects, called quantum groups. My thesis extends their result to a different, related, collection of objects.

If you don’t know anything about matrices this is as far as you can understand I’m afraid. You’ll have to be satisfied with the answer I’ve given above: that I was interested in classifying some abstract algebraic objects related to centuries-old groups.

If you have studied a first course in linear algebra, we can talk about representations a little bit. Matrices are a way of encoding transformations between linear spaces. Representation theory tries to make group structures concrete by “representing them” by matrices. The classification of representations then translates into classifying eigenspaces of various matrices. If you are aware of how ubiquitous linear algebra is in all computational sciences, you should be able to appreciate the usefulness of decomposing eigenspaces into smaller, more tractable, “atomic” pieces: these are known as “irreducible representations” in representation theory: my goal was to classify irreducible representations of objects I called alternating quiver Hecke algebras.

What is an alternating quiver Hecke algebra?

Let’s break it down into the four words separately:

• alternating: the word alternating comes from the alternating group, which is related to the symmetric group I mentioned earlier. This signifies the objects belong to a family that has been studied for a very long time.
• quiver: a quiver is a directed graph (a bunch of dots connected with arrows). Quivers are used to encode important information about the objects.
• Hecke: Hecke was a mathematician who first introduced Hecke algebras. Hecke algebras are mathematical objects closely related to symmetric groups.
• algebra: this is the type of mathematical structure I studied.

Why was your thesis result interesting?

The main result in my thesis proved the existence of a new structure in the family of alternating groups. Precisely: I showed that the (modular group algebras of) alternating groups are graded by the integers. This sheds considerable new light on a family of objects that have been studied for over a hundred years. It also directly connects them with the seemingly disparate world of quantum groups.