Fields Medalists announced in India

By Camille M. Carlisle

Web edition: August 19, 2010

British mathematician Godfrey Hardy once claimed proudly that his work could never be applied to the real world. (He was later proven wrong.) That tone doesn’t resonate in the work of the four Fields Medalists announced today at the International Congress of Mathematicians in Hyderabad, India. While theoretically beautiful, these men’s achievements all have glints of physics in them, too.

For example, Stanislav Smirnov of the University of Geneva won “for the proof of conformal invariance of percolation and the planar Ising model in statistical physics.” What that mouthful means is that Smirnov managed to prove mathematically physicists’ intuition about models of large-scale behavior determined by probabilistic, small-scale behavior. Take, for instance, water flowing through soil. As Julie Rehmeyer explains in her summary of Smirnov’s work, the water needs a clear path between dirt grains if it’s to pass through. The existence of a channel depends on the probability of a gap being in any given spot. You could attack this problem with a grid of dots to mark each spot, but calculating the probability of flow with such a model only takes you so far if you don’t know how fine the theoretical “lattice grid” should be to approximate reality. Physicists long suspected that a scaling limit existed such that finer grids approached some probability for the channel’s existence, but Smirnov actually proved it mathematically for triangular lattices in 2001.

Elon Lindenstrauss of Hebrew University in Jerusalem made his Fields-winning mark in ergodic theory, an area of mathematics developed to study dynamical systems. His work is particularly important at the quantum scale, where particles’ positions can only be known probabilistically. Lindenstrauss proved that the probability distribution in an area calculated in a classically dynamic way becomes more evenly distributed as the energy of the system goes up — and only for that particular way of calculating the area.

Cédric Villani, director of the Henri Poincaré Institute in Paris, was recognized for work with colleagues on how a system’s entropy increases with time. They discovered that, while disorder always increases, it doesn’t always go at the same rate. He also proved Soviet physicist Lev Landau’s claim that plasma approaches equilibrium not by spreading around a room like regular gas particles but because of decay in the electric field that the plasma’s ionized particles create.

Ngô Bào Châu of Princeton University earned the Fields Medal by solving mathematician Robert Langlands' Fundamental Lemma ( a “lemma” is a minor statement used as a stepping-stone for a larger proof). Châu used geometrical objects called Hitchen fibrations to recast the complex problem into a simpler form that, as Rehmeyer puts it, “created genuine understanding” and might provide the techniques necessary to fully prove Langlands’ larger conjectures on uniting seemingly unconnected fields of mathematics.

Other awards announced at the meeting were the Nevanlinna Prize, to Daniel Spielman of Yale University; the Gauss Prize, to Yves Meyer of École Normale Supérieure de Cachan, France (emeritus); and the Chern Prize to Louis Nirenberg of New York University.