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Features: Faculty Insights


Congratulations to Oscar Randal-Williams, who has been elected to the prestigious Sadleirian Professorship of Pure Mathematics in the Department of Pure Mathematics and Mathematical Statistics, with effect from January 2024.

The Sadleirian chair has a long and illustrious history. The first Sadleirian Professor, appointed in 1863 with an endowment left by Lady Mary Sadleir in 1701, was Arthur Cayley, a towering figure of nineteenth century mathematics who'll be familiar to many through the variety of mathematical objects and results that carry his name: from Cayley's theorem to the Cayley Graph. Other Sadleirian Professors have included the number theorists G.H. Hardy and Louis Mordell, and, most recently, Emmanuel Breuillard.

Randal-Williams works in algebraic and geometric topology. "These two subjects historically had a lot to do with each other but developed in different directions, and these days people who tend to do one tend not to do the other," he says. "But I like the interaction, [taking] questions about spaces that people are interested in from other parts of mathematics, like geometry, and then using the tools of algebraic topology to study them."

My main motivation is being outraged by not knowing things! And this is such an utterly basic thing, a really fundamental understanding of what we mean by symmetries of these spaces. Oscar Randal-Williams

Randal-Williams is particularly excited about his most recent work, which seeks to answer fundamental questions in topology. The most basic spaces in topology are the same ones you learned about in school – Euclidean spaces. These are the most concrete of spaces, pinned down by some number of coordinate axes like the vertical and horizontal axes of a graph you drew at school. The dimension of your space is given by the number of coordinate axes you need to map it. Randal-Williams is trying to answer the question: what are the symmetries of Euclidean spaces, particularly of higher dimensional spaces of five or more dimensions?

Smooth symmetries

The answer to this question depends on what kind of symmetries you are talking about. For example, think about the three-dimensional Euclidean space we are all familiar with two perpendicular horizontal axes (for forward-back and left-right) and a vertical axis. Then every point in that Euclidean space is written in terms of these three axes. Rotating these three axes by, say 45 degrees, around some axis of rotation leaves the space essentially unchanged. For example everything is still the same distance apart and the coordinate axes are still straight lines that are all perpendicular to each other – you might just have to slightly rewrite your descriptions of points in the space to be in terms of these rotated axes. The rotation acts as a symmetry of the Euclidean space.

Rotations, along with reflections, are the symmetries that we instinctively think of. These symmetries, and their generalisation to higher dimensions, can all be described as smooth. (Mathematically, a smooth symmetry preserves differentiability.) In fact all the smooth symmetries, after some deformation, can be expressed in terms of rotations and reflections.

The smooth symmetries of a Euclidean space define a space of symmetries. You can imagine starting at the rotation described above and moving slightly in this space of symmetries. This might take you to a rotation of slightly more or less than 45 degrees, or one where the axis of rotation is moved a little bit. "Then that's a similar rotation – it's not the same but it’s nearby to the one you had before,” says Randal-Williams. “It makes sense to think about walking about in the space of symmetries." This space of smooth symmetries of Euclidean spaces has been well understood for roughly a century – mathematicians have a description of the space in terms of its algebraic topology.

Continuous symmetries

But there's another type of symmetry called continuous symmetries that are much more difficult to understand than smooth symmetries: "You can't really think about continuous symmetries directly, it's like looking at the sun somehow!" Continuous symmetries have to send nearby points to nearby points but they don't have to be smooth, in the sense that they could, for example, send an axis to some very jagged curve.

The space of continuous symmetries includes all the smooth symmetries, but mathematicians have known since the 1970s that there are continuous symmetries that aren't smooth. But we don't have, as of yet, a description of the space of continuous symmetries of Euclidean space.

For mathematicians, including Randal-Williams, describing all the symmetries of Euclidean space, not just the smooth symmetries, is a basic question to ask. "My main motivation is being outraged by not knowing things!" he says. "And this is such an utterly basic thing, a really fundamental understanding of what we mean by symmetries of these spaces."

Euclidean spaces can be thought of as the basic case of topology, which deals more generally with manifolds – that is, anything that looks locally like a Euclidean space. "So if you can answer the question for Euclidean space you're on the way to answer much more."

Randal-Williams and his colleagues are now tantalisingly close to a complete description of the continuous symmetries for every Euclidean space of any dimension of 5 or more. "We haven't quite proved it in complete detail but it makes so much internal sense I'm confident it must be right."

Rephrasing the question

As it's so hard to consider continuous symmetries directly, Randal-Williams and his colleagues Alexander Kupers and Manuel Krannich (Krannich was a postdoc in Cambridge when this work started) instead found a way to rephrase this question. It turns out that understanding the continuous symmetries of Euclidean space is more or less interchangeable with understanding smooth symmetries of a disc (that is, symmetries that fix the boundary of the disc).

Using this new point of view, Randal-Williams and his colleagues were able to develop and find strong evidence for their conjecture: that the space of continuous symmetries of any high-dimensional Euclidean space can be expressed in terms of algebraic objects called graph complexes.

Graph complexes have only really been studied in any detail in the last 10-15 years. They are built out of finite graphs – networks with finitely many nodes and links (edges) between them – and an operation that captures all the ways of contracting an edge.

The incredible thing about these graph complexes is that they are so elementary, says Randal-Williams. The definition above is very vague, but even a rigorous mathematical definition is surprisingly easy to state. "And the same graph complexes show up in surprisingly different bits of maths. And I don't just mean the same kind of thing, I mean literally the same graph complex."

Surprising connections

The graph complexes from his work on symmetries of Euclidean spaces also arise in mathematical physics (which is where graph complexes first emerged), and are related to moduli spaces of Reimann surfaces in algebraic geometry. "It's incredible it's literally the same [graph complex] that answers these different questions. In particular it shows that those questions are in some way related, because they are both related to the same little combinatorial object." But if you were not to consider graph complexes, it would be very difficult to see how these areas of mathematics related to each other. "It really is some sort of primitive thing. It's like the prime numbers – it's a building block that shows up in lots of different buildings."

The fact that graph complexes provide the description Randal-Williams and his colleagues were looking for is quite surprising. "The connection is quite indirect. Why should the symmetries of Euclidean spaces have anything to do with finite graphs? That is not obvious at all." They are now working on the next steps to prove their conjecture by working with the world expert on graph complexes, Thomas Willwacher.

Randal-Williams' work on this conjecture illustrates his broad approach to mathematics, which draws on and contributes to many different areas. The Sadleiran Professorship was founded for research in algebra, and Randal-Williams brings together this area with other parts of pure mathematics. "I'm very pleased personally, but it is also encouraging recognition of the area of algebraic topology which is having a renaissance," he says. It seems particularly fitting that the new holder of this long-established and prestigious professorship is one of the new generation of mathematicians revolutionising our fundamental understanding of pure mathematics.