The four winners of the 2014 Fields medals – the most prestigious prizes for mathematics – were announced today, including the first female and first Latin American recipients of the 78-year-old prize.

The awarding of these and several other International Mathematical Union (IMU) prizes – typically by the head of state of the host country – is the climax of the opening ceremony of the quadrennial International Congress of Mathematicians, held this year in Seoul.

Unlike the Nobel prizes and many other modern science prizes, the financial value of the Fields medal (officially known as the International Medal for Outstanding Discoveries in Mathematics) is relatively trivial – roughly US$15,000 – but its prestige is life-changing.

The money comes from the 1932 bequest of Canadian mathematician J C Fields who ran the 1924 Mathematics Congress in Toronto. Indeed, the Canadian mint produces the medals. A friend of mine once transported them to the congress in his luggage.

According to the IMU, as was requested by Fields the man, the award intends:

to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.

Since 2003, the Abel prize has been awarded on the same scale as a Nobel prize, but the Fields medal is the prize mathematicians pay most attention to. The 2006 medallist, Australian Terry Tao, is considered special even among this august group.

The list of previous winners charts the past century’s mathematical achievements, and the four new winners make exciting additions. They include both the first woman and the first Latin American to win the prize. They also include a number theorist who many (in the know) had tipped to win.

As described in the press, the winners in alphabetic order are:

## Artur Avila

Artur Avila, a Brazilian mathematician, 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. This is typified in the classic example of a butterfly’s wings flapping leading to a change in weather hundreds of kilometres 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 evolve eventually into a stable state, or fall into a chaotic stochastic state, in which their behaviour can be described probabilistically.

## Manjul Bhargava

Canadian-American 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, such as 3x^{2} + 4xy - 5y^{2.}

Carl Friedrich Gauss, one of the greatest mathematicians of the late 18th and early 19th centuries, developed a powerful tool for analysing polynomials such as 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 – originally from Austria and currently professor at the University of Warwick – researches stochastic partial differential equations. Differential equations show up throughout mathematics, physics and engineering. They describe processes that change over time, such as 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 involve only one variable. 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:

- a 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 right and with powerful applications in the sciences and engineering.

## Maryam Mirzakhani

Iran-born Stanford professor 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 such as 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 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, such as triangles with three 90-degree angles.

Riemann surfaces can get far more complicated than the Riemann sphere. One of the major research areas in the study of these surfaces is how one Riemann surface can be smoothly deformed or smooshed 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.

The official citations for all the 2014 IMU prizes are available on the IMU site.

## And is it the Nobel?

The folk history of why there is no Nobel prize in mathematics is amusing and largely bogus. The real history is even more amusing – if you’re interested, a New York Times article from last week describes it beautifully.

Helpfully, Scientific American has even published cocktail party talking points.

Nobel or not, these four prizes celebrate the enormous depth, range and scope of modern mathematics. Both Fields and Nobel would be proud.