On 13 August 2014, at the quadrennial meeting of the International Congress of Mathematicians, this year held in Seoul, Korea, the four winners of the 2014 Fields Medals were announced by the International Mathematics Union, which administers the awards.

This year’s awardees are:

- Artur Avila, a Brazilian mathematician (the first Brazilian mathematician to win the prize) has done notable research in the study of chaos theory and dynamical systems. These areas seek to understand the behavior of systems that evolve over time in which very small changes in the initial conditions can lead to wildly varying outcomes. One well-known example is weather forecasting, where a butterfly’s flapping of wings can ultimately lead to different weather elsewhere on the planet. Avila’s principal contribution was to classify dynamical systems into those that eventually converge to a stable state and those that fall into a chaotic state which can only be described probabilistically.
- Manjul Bhargava is a Canadian-American mathematician who has focused on algebra and number theory, in particular the behavior of polynomials with integer coefficients (e.g., 3x
^{2}+ 4xy – 5y^{2}). Such polynomials were originally studied by Gauss. Bhargava, by intensely studying Gauss’ work and adding insight from geometry and algebra, extended Gauss’ tool to higher-degree polynomials. - Martin Hairer, originally from Austria but now at the University of Warwick in the U.K., studies stochastic partial differential equations. Ordinary differential equations typically only involve one time-dependent variable (e.g., the displacement of a cannonball with time), whereas partial differential equations are required to study phenomena (e.g., the earth’s climate) that involve multiple interacting variables as well as time. Hairer studied stochastic partial differential equations, which are partial differential equations that involve randomly varying variables. Such equations arise in quantum field theory and statistical mechanics, among other applications. His theoretical framework has made the study of these equations significantly more tractable.
- Maryam Mirzakhani breaks ground in two respects: she is the first woman and also the first Iran-born mathematician to win the prize. Her work focuses on the geometry of Riemann surfaces, namely surfaces where the measurement of angles and distances is different than on a normal Euclidean plane. A simple example is the Riemann sphere, which can thought of as a mapping from a sphere to the plane passing through the equator by noting the intersection of a line passing from the north pole through a point on the surface, and then continuing to intersect the plane (at some point outside the sphere). This plane, with the “point at infinity” added, is the Riemann sphere, which is useful for the study of complex analysis among other things. Riemann surfaces are more general and can be far more complicated than the Riemann sphere. Mirzakhani has contributed substantially to the study of deformations of Riemann surfaces, which have their own geometries called “moduli spaces.”

Additional details on these awards are given in an article by one of us in the Conversation, and also in articles appearing in the BBC News, the

New York Times, New Scientist, the UK Guardian and the Simons Foundation Quanta Magazine (individual articles on Avila, Bhargava, Hairer and Mirzakhani).