Fields Medals Awarded On August 19, 2010, four Fields Medals were Bourgain, Pierre-Louis Lions, Jean-Christophe Yoc- awarded at the opening ceremonies of the In- coz, and Efim Zelmanov (1994); Richard Borcherds, ternational Congress of Mathematicians (ICM) in William Timothy Gowers, Maxim Kontsevich, and Hyderabad, India. The medalists are Elon Linden- Curtis T. McMullen (1998); Laurent Lafforgue and strauss, Ngô Bao Châu, Stanislav Smirnov, and Vladimir Voevodsky (2002); and Andrei Okounkov, Cédric Villani. Grigori Perelman, (medal declined), Terence Tao, The Fields Medals are given every four years by and Wendelin Werner (2006). the International Mathematical Union (IMU). Al- though there is no formal age limit for recipients, Elon Lindenstrauss the medals have traditionally been presented to Citation: “For his results on measure rigidity in ergo- mathematicians under forty years of age as an dic theory and their applications to number theory.” encouragement to future achievement. The medal Elon Lindenstrauss is named after the Canadian mathematician John has developed extraor- Charles Fields (1863–1932), who organized the dinarily powerful theo- 1924 ICM in Toronto. At a 1931 meeting of the retical tools in ergodic Committee of the International Congress, chaired theory, a field of mathe- by Fields, it was decided that funds left over from matics initially developed the Toronto ICM “should be set apart for two med- to understand celestial als to be awarded in connection with successive mechanics. He then used International Mathematical Congresses.” In outlin- them, together with his ing the rules for awarding the medals, Fields speci- deep understanding of fied that the medals “should be of a character as ergodic theory, to solve Photographs courtesy of ICM/IMU. purely international and impersonal as possible.” a series of striking prob- During the 1960s, in light of the great expansion lems in areas of mathe- of mathematics research, the possible number of matics that are seemingly medals to be awarded was increased from two to Elon Lindenstrauss far afield. His methods four. Today the Fields Medal is recognized as the are expected to continue world’s highest honor in mathematics. to yield rich insights throughout mathematics for Previous recipients of the Fields Medal are: decades to come. Lars V. Ahlfors and Jesse Douglas (1936); Laurent Ergodic theory studies dynamical systems, Schwartz and Atle Selberg (1950); Kunihiko Kod- which are simply mathematical rules that describe aira and Jean-Pierre Serre (1954); Klaus Roth and how a system changes over time. So, for example, René Thom (1958); Lars Hörmander and John W. a dynamical system might describe a billiard ball Milnor (1962); Michael F. Atiyah, Paul J. Cohen, Al- ricocheting around a frictionless, pocketless bil- exander Grothendieck, and Stephen Smale (1966); liard table. The ball will travel in a straight line until Alan Baker, Heisuke Hironaka, Sergei Novikov, and it hits the side of the table, which it will bounce off John G. Thompson (1970); Enrico Bombieri and of as if from a mirror. If the table is rectangular, David Mumford (1974); Pierre R. Deligne, Charles this dynamical system is pretty simple and predict- Fefferman, Grigorii A. Margulis, and Daniel G. able, because a ball sent in any direction will end Quillen (1978); Alain Connes, William P. Thur- up bouncing off each of the four walls at a consis- tent angle. But suppose, on the other hand, that the ston, and Shing-Tung Yau (1982); Simon K. Don- billiard table has rounded ends like a stadium. In aldson, Gerd Faltings, and Michael H. Freedman that case, a ball from almost any starting position (1986); Vladimir Drinfel’d, Vaughan F. R. Jones, headed in almost any direction will shoot all over Shigefumi Mori, and Edward Witten (1990); Jean DECEMBER 2010 NOTICES OF THE AMS 1459 the entire stadium at endlessly varying angles. Then Lindenstrauss brought his ergodic theory Systems with this kind of complicated behavior tools to the problem, in joint work with Manfred are called “ergodic”. Einsiedler and Anatole Katok. Ergodic theory might The way that mathematicians pin down this seem an odd choice for a problem that does not in- notion that the trajectories spread out all over the volve dynamical systems or time, but such unlikely space is through the notion of “measure invari- pairings are sometimes the most powerful. Here’s ance”. A measure can be thought of as a more flex- one way of reformulating Littlewood’s problem to ible way to compute area, and having an invariant see a connection: First imagine a unit square, and measure essentially ensures that if two regions of glue the top edge to the bottom edge to make a the space in some sense have equal areas, points cylinder. Now glue the right edge to the left edge will travel into them the same percentage of the and you’ll get a shape called a torus that looks like time. By contrast, in the rectangular table (which a donut. You can roll up the entire coordinate plane of course is not ergodic), the center will get very to this same shape by gluing any point (x, y) to the little traffic in most directions. point whose x-coordinate is the fractional part of In many dynamical systems, there is more than x and whose y-coordinate is the fractional part of one invariant measure, that is, more than one y. This torus is the space of our dynamical system. way of computing area for which almost all the We can then define a transformation by taking any trajectories will go into equal areas equally often. point (x, y) to another point (x+α, y+β). If α and In fact, there are often infinitely many invariant β are irrational (or more precisely, not rationally measures. What Lindenstrauss showed, however, related), this dynamical system will be ergodic. The is that in certain circumstances, there can be only Littlewood conjecture then becomes the claim that a very few invariant measures. This turns out to you can make these trajectories suitably close to be an extremely powerful tool, a kind of hammer the origin by applying the transformation enough that can break hard problems open. times. The number of times you apply the transfor- Lindenstrauss then adroitly wielded his ham- mation becomes the denominator of the fractions mer to crack some hard problems indeed. One approximating α and β. example of this is in an area called “Diophantine Using a reformulation of the Littlewood con- approximations”, which is about finding rational jecture in terms of a more complex dynamical numbers that are usefully close to irrational ones. system, the team made a huge step of progress Pi, for example, can be approximated pretty well as on the conjecture: They showed that if there are 22/7. The rational number 179/57 is a bit closer, any pairs of numbers for which the conjecture is but because its denominator is so much larger, it’s false, there are only a very few of them, a negligible not as convenient an approximation. In the early portion of them all. nineteenth century, the German mathematician Another example of the power of Linden- Johann Dirichlet proposed one possible standard strauss’s work is his proof of the first nontrivial for judging the quality of an approximation: The case of the arithmetic quantum unique ergodicity imprecision of a rational approximation p/q should conjecture. Ergodic systems come up frequently be less than 1/(q 2). He then went on to show, in in physics, because as soon as you have three a not very difficult proof, that there are infinitely bodies interacting, for example, the system starts many approximations to any irrational number to behave in a somewhat ergodic fashion. But if that meet this standard. (To put this in formula those interactions happen at the quantum scale, form, he showed that for any real number α, there you cannot describe them with the ordinary are infinitely many integers p and q such that tools of ergodic theory, because quantum theory |α−p/q| < 1/q 2.) does not allow for well-defined paths of points Eighty years ago the British mathematician at well-defined positions; instead, you can only John Edensor Littlewood proposed an analogue to consider the probability that a point will exist in a Dirichlet’s statement to approximate two irrational particular position at a particular time. Analyzing numbers at once: It should be possible, he figured, such systems mathematically has proven extraor- to find approximations p/q to α and r/q to β so dinarily difficult, and physicists have had to rely that the product of the imprecision of the two on numerical simulations alone, without a firm approximations would be as small as you please. mathematical underpinning. (In formula form, the claim is that for any real The quantum unique ergodicity conjecture says, numbers α and β and any tiny positive quantity roughly, that if you calculate area using the mea- ε you like, there will be approximations p/q to α sure that is natural in classical dynamics, then as and r/q to β so that |α−p/q|×|β−r/q|<ε/q3.) He gave the energy of the system goes up, this probability the problem to his graduate students, thinking it distribution becomes more evenly distributed over should not be that much harder than Dirichlet’s the space. Furthermore, this measure is the only proof. But the Littlewood conjecture turned out to one for which that is true. Lindenstrauss was able be extraordinarily difficult, and until recently, no to prove this in an arithmetic context for particular substantial progress had been made on it. kinds of dynamical systems, creating one of the 1460 NOTICES OF THE AMS VOLUME 57, NUMBER 11 first major, rigorous advances in the theory of The full realization of Langlands’s program would quantum chaos.