The four winners of the Fields Medal for 2018 have been announced. The Fields Medal is awarded every 4 years at an international gathering of mathematicians and is considered the Nobel Prize for Mathematics. However, there is one key difference: recipients must be 40 years old or younger.

This years recipients were announced on Wednesday 1st August at the International Congress of Mathematicians in Rio de Janeiro. They are:

Caucher Birker, 40, of the University of Cambridge in England: “for his proof of the boundedness of Fano varieties and for contributions to the minimal model program.”

Allesio Figalli, 34, of the Swiss Federal Institute of Technology in Zurich: “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability.”

Akshay Venkatesh, 36, of the Institute for Advanced Study in Princeton and Stanford University in California: “for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.” Click here for more information.

Peter Scholze, 30 (one of the youngest recipients), of the University of Bonn in Germany: “for transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to galois representations and for the development of new cohomology theories.” Click here for more information.

Polya Prize: This was awarded to Professor Alex Wilkie FRS from the University of Oxford due to his contributions to model theory and its connections to real analytic geometry.

Senior Whitehead Prize: Professor Peter Cameron, from the University of St Andrews, was awarded this prize for his research on combinatorics and group theory.

Senior Anne Bennett Prize: Awarded to Professor Alison Etheridge FRS from the University of Oxford for her “research on measure-valued stochastic processes and applications to population biology” as well as outstanding leadership.

Naylor Prize and Lectureship: This was given to Professor John Robert King from the University of Nottingham due to profound contributions to the theory of non-linear PDEs and applied mathematical modelling.

Berwick Prize: This was awarded to Kevin Costello of the Perimeter Institute in Canada for his paper entitled The partition function of a topological field theory (published in the Journal of Topology in 2009). In this paper Costello “characterises the function as the unique solution of a master equation in a Fock space.”

Whitehead Prize:

Dr Julia Gog (University of Cambridge) for her research on the mathematical understanding of disease dynamics, in particular influenza.

Dr András Máthé (Univeristy of Warwick) due to his insights into problems in the fields of geometric measure theory, combinatorics and real analysis.

Ashley Montanaro (University of Bristol) for her contributions to quantum computation and quantum information theory.

Dr Oscar Randal-Williams (University of Cambridge) due to his contributions to algebraic topology, in particular the study of moduli spaces of manifolds.

Dr Jack Thorne (University of Cambridge) for research in number theory, in particular the Langlands program.

Professor Michael Wemyss (University of Glasgow) for his “applications of algebraic and homological techniques to algebraic geometry.”

When I read the news that Maryam Mirzakhani had sadly passed away with breast cancer aged 40 I was honestly shocked. I remember finding out that she was the first women to win the Fields Medal in 2014 and feeling a huge sense of pride that we had achieved such a big milestone in mathematics – a mostly male dominated subject.

Professor at Stanford Universtiy, Mirzakhani was awarded the notorious Field’s Medal for her work on complex geometry and dynamic systems. She specialised in areas of theoretical mathematics that “read like a foreign language by those outside of mathematics” such as moduli spaces, Teichmüller thoery, hyperbolic geometry, Ergodic theory and symplectic geometry. By mastering these fields, Mirzakhani could describe the geometric and dynamic complexities of curved surfaces, spheres, donut shapes and even amoebas in a huge amount of detail. Furthermore, her work had implications in a vast amount of fields, ranging from cryptography to the physics of how the universe was created.

Moduli Spaces

Moduli Spaces can be thought of as geometric solutions to geometric classification problems. In broad terms, a moduli problem consists of three main categories:

Objects: which geometric objects do we want to parametrise?

Equivalences: when do we identify two objects as being isomorphic?

Teichmüller theory, which brings together an array of fundamental ideas from different mathematical fields (including complex analysis, hyperbolic geometry, differential geometry, etc), is concerned with the Teichmüller space.

To get an short introduction to Teichmüller theory, click here.

Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry, where the parallel postulate of Euclidean geometry is replaced with:

“For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.” – Wikipedia

Ergodic Theory

Ergodic theory was initially developed to solve problems in statistical physics and is a branch of mathematics that studies “dynamical systems with an invariant measure”. An invariant measure is a measure that is preserved by some function.

Symplectic Geometry

Symplectic Geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. These are differentiable manifolds that have a closed, non-degenerate 2-form.

“Mirzakhani once described her work as ‘like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out’.”

– Guardian

Mirzakhani will be remembered not only for her extraordinary work, but also as being an inspiration to thousands of women to pursue maths and science.