I met with Dr Gwyn Bellamy on a sunny morning in April. With sharp features and an attentive eye, he winds his way through the labyrinthine corridors of one of the Faculty of Science’s buildings. He speaks quickly, with a noticeable Welsh accent, and peppers our conversation with the occasional “crikey”.

Dr Bellamy is a Senior Lecturer at the University of Glasgow, and is among the first visitors through SMRI’s International Visitor Program. Hailing from a small village on the Welsh coast, the school-aged Bellamy made the decision of where to study maths on a cattle showground, at a university careers fair in the village of Carmarthen.

There was this old professor guy – lots of white hair, big beard, padded jacket. He was very keen to tell me the University of Warwick was a lovely place to come. When I told him I wanted to study mathematics, he said,

Ooh, well mathematics is very hard at Warwick. What are your grades like?I thought I had done reasonably well in school, but he opened his eyes wide when I told him.

Oh, right, I see. Well, you’ll have to get someone to whip you into shape if you want to come to Warwick.That really annoyed me – I thought I had done not too badly – and so I think I went there almost out of spite.

Dr Gwyn Bellamy

Proceeding from Warwick to Edinburgh and eventually to Glasgow, Dr Bellamy has made a career studying the interplay between geometry and algebra, in particular in studying “resolutions of symplectic singularities”.

The idea arises from an attempt to model some counter-intuitive phenomena in quantum physics. If I place a marble on the table and ask you to measure both its mass and its diameter, you wouldn’t think much about whether you first put it on the scale or took out the calipers. This is because we expect to get the same answers regardless of the order in which we take the measurements: we say that these measurements *commute*.

When scientists first started investigating the world of the very small, they were surprised to find this common sense property doesn’t always hold. Max Born found that the acts of measuring a particle’s position and momentum did not commute; in other words, measuring its position and then momentum would yield a different result than measuring its momentum and then position. This fact formed the basis for Heisenberg’s famous uncertainty principle.

“Resolutions of symplectic singularities” provide one way we can make this notion of non-commutativity more tractable: these are spaces in which measurements don’t necessarily commute, but don’t fail too badly. You may not find that your particle is exactly where you thought it would be, but it won’t wind up halfway across the universe.

It turns out that the study of these spaces also gives a powerful tool for addressing longstanding problems in algebra and representation theory called the Kazhdan–Lusztig conjectures.

The idea would be to take a symplectic singularity, plug it into a general machine, turn the handle, and get an analogue of the Kazhdan–Lusztig conjectures. Unfortunately we don’t have the geometric tools to do this at the moment.

Dr Gwyn Bellamy

Dr Bellamy came to SMRI to visit Dr Oded Yacobi, Senior Lecturer in the School of Mathematics and Statistics at the University of Sydney. Dr Yacobi is an expert in “Coulomb branches”, which arise in quantum field theory. Though physicists have been using Coulomb branches for years, they have only recently been given a rigorous mathematical definition. This gives mathematicians a huge new class of examples to work with.

Bizarrely, it’s still a conjecture whether these Coulomb branches are symplectic singularities. Everyone assumes they are, but it seems quite difficult to actually show. It would be fantastic to prove that they are.

Dr Gwyn Bellamy

Dr Bellamy described to me his favourite concept in elementary mathematics: the three-dimensional shapes known as Platonic solids. The ancient Greek philosopher Plato thought they made up the four classical elements: the tetrahedron for fire, the cube for earth, the octahedron for air, and the icosahedron for water. Aristotle later added the dodecahedron for æther, making five Platonic solids.

Although modern science has embraced atoms as the more likely building blocks of matter, Platonic solids turned out to fascinate mathematicians for other reasons. It can be proved using not much more than high school maths that they are the only regular solids you can make using identical polygons for the faces. Dr Bellamy’s office in Glasgow has a modelling set, with polygons you can clip together to form Platonic solids. “They just seem to show up everywhere in mathematics when you look deeply enough.”

From Coulomb branches to Platonic solids, Dr Bellamy loves exploring the rich connections in mathematics.