Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Torus

      Moduli spaces: A journey into the world of shapes

      3 July, 2024

      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.

      INI logo

       

      Read more about...
      INI
      topology
      geometry
      surface
      Riemann surface
      • Log in or register to post comments

      Read more about...

      INI
      topology
      geometry
      surface
      Riemann surface
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms