# Fields Medal 2010

Mathematics, Research

The Fields Medal is often described as ‘the Nobel Prize for mathematics’, although the Abel Prize would come closer.  The Fields Medal was first awarded in 1936, and is awarded every four years, to two, three or four mathematicians under the age of 40.  The medal is awarded at the International Congress of the International Mathematical Union (IMU).

The 2010 winners are:
• Elon Lindenstrauss
• Ngô Bảo Châu
• Stanislav Smirnov
• Cédric Villani

The following descriptions of their research are from the International Congress of Mathematicians website.

Elon Lindenstrauss

“For his results on measure rigidity in ergodic theory, and their applications to number theory.”

Elon Lindenstrauss has developed extraordinarily powerful theoretical tools in ergodic theory, a field of mathematics initially developed to understand celestial mechanics. He then used them, together with his deep understanding of ergodic theory, to solve a series of striking problems in areas of mathematics that are seemingly far afield. His methods are expected to continue to yield rich insights throughout mathematics for decades to come.

Ergodic theory studies dynamical systems, which are simply mathematical rules that describe how a system changes over time. So, for example, a dynamical system might describe a billiard ball ricocheting around a frictionless, pocketless billiard table. The ball will travel in a straight line until it hits the side of the table, which it will bounce off of as if from a mirror. If the table is rectangular, this dynamical system is pretty simple and predictable, because a ball sent any direction will end up bouncing off each of the four walls at a consistent angle. But suppose, on the other hand, that the billiard table has rounded ends like a stadium. In that case, a ball from almost any starting position headed in almost any direction will shoot all over the entire stadium at endlessly varying angles.  Systems with this kind of complicated behavior are called “ergodic.”

continued.

Ngô Bảo Châu

“For his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.”

Ngô Bao Châu removed one of the great impediments to a grand, decades-long program to uncover hidden connections between seemingly disparate areas of mathematics. In doing so, he provided a solid foundation for a large body of theory and developed techniques that are likely to unleash a flood of new results.

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Stanislav Smirnov

“For the proof of conformal invariance of percolation and the planar Ising model in statistical physics.”

Stanislav Smirnov has put a firm mathematical foundation under a burgeoning area of mathematical physics. He gave elegant proofs of two long-standing, fundamental conjectures in statistical physics, finding surprising symmetries in mathematical models of physical phenomena.

Though Smirnov’s work is highly theoretical, it relates to some surprisingly practical questions. For instance, when can water flow through soil and when is it blocked? For it to flow, small scale pores in the soil must link up to provide a continuous channel from one place to another. This is a classic question in statistical physics, because the large-scale behavior of this system (whether the water can flow through a continuous channel of pores) is determined by its small-scale, probabilistic behavior (the chance that at any given spot in the soil, there will be a pore).

It’s also a natural question to model mathematically. Imagine each spot in the soil as lying on a grid or lattice, and color the spot blue if water can flow and yellow if it can’t. Determine the color of each spot by the toss of a coin (heads for yellow, tails for blue), using a coin that might be weighted rather than fair. If a path of blue spots crosses from one side of a rectangle to the other, the water can pass from one side to the other.

Such “percolation models” behave in a remarkable way. For extreme values, the behavior is as you might expect: If the coin is heavily weighted against blue, the water almost certainly won’t flow, and if it’s heavily weighted toward blue, the water almost certainly will. But the probability of flow doesn’t change evenly as the percentage of blue spots increases. Instead, the water is almost certainly going to be blocked until the percentage of blue spots reaches some threshold value, and once it does, the probability that the water will flow starts surging upward. This threshold is called the “critical point.” Abrupt change of behavior like this is a bit like what happens to water as it heats: suddenly, at a critical temperature, the water boils. For that reason, this phenomenon is commonly called a phase transition.

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Cédric Villani

“For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.”

Cédric Villani has provided a deep mathematical understanding of a variety of physical phenomena. At the center of much of his work is his profound mathematical interpretation of the physical concept of entropy, which he has applied to solve major problems inspired by physics. Furthermore, his results have fed back into mathematics, enriching both fields through the connection.

Villani began his mathematical career by re-examining one of the most shocking and controversial theories of 19th century physics. In 1872, Ludwig Boltzmann studied what happens when the stopper is removed on a gas-filled beaker and the gas spreads around the room. Boltzmann explained the process by calculating the probability that a molecule of gas would be in a particular spot with a particular velocity at any particular moment – before the atomic theory of matter was widely accepted. Even more shockingly, though, his equation created an arrow of time.

The issue was this: When molecules bounce off each other, their interactions are regulated by Newton’s laws, which are all perfectly time-reversible; that is, in principle, we could stop time, send all the molecules back in the direction they’d come from, and they would zip right back into the beaker. But Boltzmann’s equation is not time-reversible. The molecules almost always go from a state of greater order (e.g., enclosed in the beaker) to less order (e.g., spread around the room). Or, more technically, entropy increases.

Over the next decades, physicists reconciled themselves to entropy’s emergence from time-reversible laws, and indeed, entropy became a key tool in physics, probability theory, and information theory. A key question remained unanswered, though: How quickly does entropy increase? Experiments and numerical simulations could provide rough estimates, but no deep understanding of the process existed.

Villani, together with his collaborators Giuseppe Toscani and Laurent Desvillettes, developed the mathematical underpinnings needed to get a rigorous answer, even when the gas starts from a highly ordered state that has a long way to go to reach its disordered, equilibrium state. His discovery had a completely unexpected implication: though entropy always increases, sometimes it does so faster and sometimes slower. Furthermore, his work revealed connections between entropy and apparently unrelated areas of mathematics, such as Korn’s inequality from elasticity theory.

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