The Fields Medal is widely considered the most prestigious award in mathematics, and the 2014 Fields Medalists were just announced.

The Fields Medal is given to between two and four mathematicians under age 40 at the International Congress of Mathematicians, held once every four years. According to the International Mathematical Union, the body that awards the Fields Medal, the award intends “to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.”

It is usually awarded for mathematical research that solves or extends problems that have vexed mathematicians for decades or centuries, or research that greatly expands on or even creates new areas of mathematical thought.

Here are the four 2014 Fields Medalists:

## Artur Avila

Artur Avila is a Brazilian mathematician who has contributed to a number of fields. Some of his most notable research is in the study of chaos theory and dynamical systems. These areas seek to understand the behaviour of systems that evolve over time in which very small changes in initial conditions can lead to wildly varying outcomes, such as weather patterns, as typified in the classic example of a butterfly’s wings flapping leading to a change in weather hundreds of miles away.

One of Avila’s major contributions to this field of study was in clarifying that a certain broad class of dynamical systems fall into one of two categories. They either eventually evolve into a stable state, or fall into a chaotic stochastic state, in which their behaviour can be described probabilistically.

## Manjul Bhargava

Manjul Bhargava‘s research is focused on number theory and algebra. One of the basic subjects in algebraic number theory is the behaviour of polynomials with integer coefficients, like 3x^{2} + 4xy -5y^{2}.

Carl Friederich Gauss, one of the greatest mathematicians of the late 18th and early 19th centuries, developed a powerful tool for analysing polynomials like the one above, where the variables are all raised to at most the second power.

Bhargava, by intensely studying Gauss’ work and adding to it an impressive level of geometric and algebraic insight, was able to extend Gauss’ tool to higher degree polynomials in which we raise the variables to higher powers than two. This work vastly expands the ability of number theorists to study these fundamental mathematical objects.

## Martin Hairer

Martin Hairer researches stochastic partial differential equations. Differential equations show up throughout mathematics, physics, and engineering. They describe processes that change over time, like the movement of a shell shot from a cannon, or the price of a stock or bond.

Differential equations come in a variety of flavours. Ordinary differential equations are equations that only have one variable involved. The motion of a cannonball, for example, can be modelled with a simple ordinary differential equation in which the only variable is the time since the cannon was fired.

Partial differential equations involve processes that depend on multiple variables. In many physical settings, both time and the current position of an object are needed to determine the future trajectory of the object. These describe a much wider variety of processes in the world, and are generally much harder to work with than one-variable ordinary equations.

Differential equations can also be either deterministic or stochastic. The cannonball’s movement, or the movement of a satellite orbiting earth, are deterministic: Outside of measurement error, once we’ve solved the equation, we have no doubt about where the cannonball or satellite will be at a given point in time. Stochastic equations have a random element involved. The motion of sugar grains stirred in a cup of coffee, or a stock’s price at a given moment in time are both best described by models that have an element of noise or randomness.

Stochastic partial differential equations — equations that have multiple input variables and random noise elements — have traditionally been very difficult for mathematicians to work with. Hairer developed a new theoretical framework that makes these equations far more tractable, opening the door to being able to solve a number of equations with both large amounts of mathematical interest in their own rights, and with powerful applications in the sciences and engineering.

## Maryam Mirzakhani

Maryam Mirzakhani‘s work focuses on the geometry of Riemann surfaces. Riemann surfaces are a classic type of non-Euclidean geometry: while a Riemann surface still has two dimensions like a plane, and we can still define lines, angles, and curves on the surface, the way that the measurement of angles and distances will come out can be very different from what happens on a normal Euclidean plane.

A basic example of this that we’ve looked at before is the Riemann sphere: a version of a sphere in which we still have a notion of lines and angles, but where strange things can happen, like triangles with three 90-degree angles.

Riemann surfaces can get far more complicated than the Riemann sphere, and one of the major research areas in the study of these surfaces is how one Riemann surface can be smoothly deformed or smushed into another surface. These deformations themselves have their own strange geometries, called “moduli spaces,” and Mirzakhani has contributed several interesting results in understanding these mysterious spaces.

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