What would you think if the nice café latte in your cup suddenly separated itself out into one half containing just milk and the other containing just coffee? Probably that you, or the world, have just gone crazy. There is, perhaps, a theoretical chance that after stirring the coffee all the swirling atoms in your cup just happen to find themselves in the right place for this to occur, but this chance is astronomically small.

Cédric Villani, Institut Henri Poincaré
Fields medallist 2010.

The fact that such spontaneous separation never occurs in practice is an illustration of a deep physical law: it says that the entropy of a system, a measure of its complexity, almost always increases as time passes. When you first pour the milk into your coffee, for a split second milk and coffee will be neatly separated, but soon the milk disperses. The mixture of milk and coffee becomes more and more complex, until it reaches an equilibrium when both are completely mixed up and complexity is at its peak.

In the late 19th century the physicist Ludwig Boltzmann studied this phenomenon, looking at what happens when a gas is released into a room from a bottle. He came up with an equation describing the evolution of this process — the change over time — in terms of how individual atoms collide and other forces acting on the gas. His calculations showed that indeed entropy doesn't decrease. The atoms of gas start out in an ordered state — all sitting in the bottle — and end up in a state of maximum complexity, dispersed throughout the room.

Intriguingly, this result meant that there is what physicists call an arrow of time, something that isn't inherent in classical...

Suppose you throw an equal number of white and black balls into a rectangular box which is, say, 30 balls long, 10 balls wide and is now 5 layers deep in balls. What it the probability that you have a run of touching white balls from one end of the box to the other?

Stanislav Smirnov

This question, asked all the way back in 1894 in the first issue of the American Mathematical Monthly, turned out be far from simple. In fact it appears to be the earliest reference to the rich mathematical field of percolation theory, according to Harry Keston, who told the International Congress of Mathematicians about Stanislav Smirnov's work in this area that lead to Smirnov winning the 2010 Fields Medal.

Just as the name conjures up the image of water percolating through soil or porous rock, percolation theory models this mathematically as a liquid flowing through a lattice of pipes. The points where the pipes join (mathematically known as the vertices of the lattice) are either blocked, stopping the flow of liquid, or open, allowing the liquid to flow through. You can imagine that if each vertex has a high probability p of being open (and the lattice is very porous like sand) then we can be fairly certain that the liquid will flow through the lattice of pipes. And if the probability p that each vertex is open is low (and the lattice is impermeable like hard clay) then we can be fairly sure the liquid is going to get stuck and not make it the whole way through.

It turns out that there is a particular critical probability for the vertices being open, p_c, that determines exactly when a liquid can percolate across the lattice. If the probability that the...

Elon Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in number theory.

Elon Lindenstrauss, Princeton University
Fields medallist 2010.

As the name suggests, number theory studies the basic properties of numbers. The whole numbers 1, 2, 3, etc are probably the first thing that spring to mind when you think about numbers. Close to follow are the rational numbers: these are the fractions, numbers of the form , where and are both whole numbers. But there also irrational numbers, which can't be written as fractions. An example is the number : some people write it as 22/7, but that's just an approximation: it's close to...

Results in mathematics come in several flavours — theorems are the big important results, conjectures will be important results one day when they are proved, and lemmas are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?

It turn's out that Ngô's lemma, formulated in 1970 as part of the famous Langlands programme, wasn't so small after all. And after an enormous amount of mathematical theory came to rely on this unproven lemma, it got a promotion and became known as the Fundamental Lemma.

In the 1970's the mathematician Robert Langland had a grand vision that could bring together the seemingly unrelated fields of group theory, number theory, representation theory and algebraic geometry. Langlands work laid out a mathematical map connecting these diverse areas of mathematics which has lead to a large area of research known as the Langlands program. One of the most important tools in this work is the trace formula, an equation which allows arithmetic information to be calculated from geometric information, itself linking together the disparate concepts of the continuous (a property of things that can be divided into infinitisemal parts, which include geometric objects such as lines, surfaces and the the three-dimensional space we live in) and the discrete (describing things that come in whole indivisible parts, such as the whole numbers which are studied in number theory).

However in order for the trace formula to be applied in any useful way in the Langlands program, an apparently relatively simple condition, that two complicated sums were equal, needed to hold. Langlands and others assumed this...