# Faculty of Mathematics

## Dr Oscar Randal-Williams, from the Department of Pure Mathematics and Mathematical Statistics, has established himself as a clear leader in the field of algebraic topology – the area of maths that uses algebra to investigate geometry and topology. In recognition of his work he has received a Philip Leverhulme Prize, one of the three Cambridge winners in 2017.

Each year 30 outstanding researchers whose work has already attracted international recognition and whose future career is exceptionally promising are awarded Philip Leverhulme Prizes. The prizes are each worth £100,000, which can be used over two or three years to advance the winners' research.

Randal-Williams works on moduli spaces - where rather than considering one mathematical object at a time, you instead consider the space of all such mathematical objects. "In the old days, people used to study mathematical objects one at a time - you take one triangle in a plane and ask questions about that particular triangle. But these days we prefer to put them all into one object and think about the space of all possible triangles." This would form a moduli space, where each point in the space corresponds to a particular triangle. Moving on a path through the moduli space corresponds to starting with a particular triangle, and then changing the angles and lengths of the triangle.

Such moduli spaces can be studied for their own properties. For example, what is the shape of that space? Does this space have a hole in it – like a doughnut does - or is it more like the surface of a ball? Does it have higher-dimensional holes? How many holes? Because these spaces are often hard to visualise, algebra can offer a more fruitful way to describe and answer such topological questions.

## A new tool for moduli spaces

Randal-Williams has been working with his colleagues Soren Galatius and Alexander Kupers on a new tool that could shed new light on certain types of moduli spaces. Their work makes use of an extra structure (called an E2-structure) that has long been known but has never before been used in this way. This simplest example of this structure comes from considering the space of points on a flat plane, but the structure arises in many different moduli spaces mathematicians are interested in. Randal-Williams has explored what other settings gives rise to moduli spaces with the same structure, and developed new ways for using this common structure.

"This is a new tool that can be used in different settings within pure mathematics." The work is still in progress, and he, Galatius and Kupers have written a foundational document which will allow researchers from different areas to use the new tool to explore this connection. For Randal-Williams the Leverhulme Prize not only rewards work that has already been done: "I think, in a sense, these prizes are awarded for work that's yet to come."

You can read more about other recent Cambridge winners of this prize here.