Artin groups and diagram algebras via topology

Mathematics describes in the simplest and purest terms the objects around us, and the world we live in. The real world is often more complex that we can model mathematically, but that does not mean that studying the underlying mathematics is not important. Take the mathematical concept of the sphere: we can never build a perfect sphere (every real life model will have microscopic imperfections), but understanding spheres can help us understand the earth. After all, the earth is a sphere which spins (a mathematical symmetry) and thus gives us night and day - arguably the most fundamental symmetry. Mathematics surrounds us every day: swings and roundabouts, famous art and architecture, the layout of robots on factory floors, symmetries in chemical compounds, and hair braids are all rich examples of mathematical phenomena.

In this project I will study 'algebraic objects' using 'topological methods'. Let's look at the methods first. "Topology" is a field of mathematics which studies the underlying 'shape' of an object, no matter how much you bend, squash, or stretch it. For example, a circle has one hole, and no matter how much you squash it or stretch it, the hole is never changing. Similarly, a figure 8 has two holes. 'Holes' in an object are described by the topological notion of "homology", which is potentially the most successful tool topology has to offer, and an example of something mathematicians call a "topological invariant". It can, for example, tell us that a circle and a figure 8 are inherently 'different' shapes. Rather than taking a concrete object like a circle to begin with, I will take an algebraic object, and build a concrete object which 'resembles' the algebraic object in some sense. Computing topological invariants of this concrete object can then provide information about the original algebraic object.

So let's return to our algebraic objects, of which there are 2 types. The first type of algebraic objects I will study in this proposal are "Artin groups". In mathematics, a "symmetry" of an object is a transformation that leaves the object 'unchanged': real life examples are shuffling a stack of papers, rotating a roundabout, or braiding your hair. The symmetries of an object are packaged up into an algebraic object called a "group". Artin groups are a broad family of groups with the simplest example being the braid group - one can think of the braid group as the mathematical embodiment of all the ways to braid your hair using only a specific number of strands. Artin groups are very simple to define mathematically, yet very mysterious: even after over 50 years of work, many fundamental questions remain unanswered. I aim to build concrete topological objects to study Artin groups, and answer some of these fundamental questions.

The second type of algebraic objects I will study in this proposal are "diagram algebras". Consider 2n holes in the ground, and n moles which have to pop out of one hole and make their way to a free hole. If we consider the paths these moles create we get a diagram belonging to some of the diagram algebras I will study: allowing the moles to cross each others' paths gives us the "Brauer algebra", and if we don't allow them to cross we get the "Temperley-Lieb algebra". Diagram algebras have strong connections with physics: replacing our moles with particles, gives a physicists "connection diagram". I aim to study the homology of these algebras individually, and also to adapt a general framework for studying homology to include the homology of these algebras.

Studying these algebraic objects using methods from topology will result in interesting pure mathematics, and add to our collective understanding of the pure phenomena underlying the world we live in. The links that generalised braiding and diagram algebras have to other areas of mathematics and science also means that this project will have a knock-on effect that will benefit research across the sciences.