Think of a geometric shape, any shape. Lines, circles and triangles come to mind. Going up a dimension you get surfaces. The perfect sphere is one example. Others are a flat piece of paper, or the surface of a doughnut.

How many different surfaces are there and what do they all look like? That question seems impossible to answer. But mathematicians are good at classification, and they have a way of dealing with multitudes. In this series of articles we give an intuitive introduction to *moduli spaces*, which give you a way of understanding all surfaces of a certain type in one go. We do this using the example of *Riemann surfaces*, named after the 19th century mathematician Bernhard Riemann, that are shaped like a doughnut. The articles visit some high-powered mathematics, but we have made them as accessible to the mathematically untrained as possible (with links to more technical explanations for those who want to know more).

The articles were inspired by the research programme *New equivariant methods in algebraic and differential geometry* which took place at the Isaac Newton Institute for Mathematical Sciences in Cambridge between January and June 2024. The articles were produced in collaboration with Ruadhaí Dervan and Victoria Hoskins, who co-organised the programme.

Moduli spaces: Introducing Riemann surfaces — This article gives an introduction to Riemann surfaces and to what we might mean by saying that two Riemann surfaces are the same.

Moduli spaces: What type of Riemann tori are there? — This article looks at Riemann surfaces that are shaped like doughnuts (tori) and introduces a neat way of deciding whether two of them are the same. It involves unfolding and flattering out the tori in a clever way.

Moduli spaces: Exploring the torus space — This article explores a clever way of describing the entire universe of Riemann tori in one go, and why this might be an interesting thing to do. It involves creating a pretty pattern on the plane and a beautiful connection to group theory.

### Background reading

Maths in a minute: Topology — In topology classifying surfaces is easy. It all comes down to the number of holes!

Maths in a minute: Complex numbers — Riemann surfaces are intimately related to the complex plane, which is made up of a family of numbers called complex numbers. Find out more about these beasts in this brief introduction.

*This content was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI). The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. The Newton Gateway is the impact initiative of the INI, which engages with users of mathematics. You can find all the content from the collaboration here. *