I’ve always wondered what The Other Guys did in their laudatios for the 2006 Fields. I’ve seen Charlie Fefferman on Terry Tao; what about the rest?

(Tangent: Charlie and Terry, I will never tire of repeating this, are like brothers. Not only were they both advised by Elias Stein at Princeton for graduate school, they were both Fields Medalists, in fact two of the top-five youngest ever, both truly world-class child prodigies with near-perfect accelerated learning trajectories who got their PhDs before age 21 and became the youngest full professors at a major American research institution and later got labeled “the best mathematician in the world” by the media at some point in their careers. A closer parallel does not exist. Yes, that’s a dare; prove me wrong.)

Today I realized I had been stupid. The URL for Charlie on Terry goes like so: http://www.icm2006.org/proceedings/Vol_I/7.pdf. Obviously you could try removing everything after ‘org’! So I did, and that was how I found the old ICM page, clearly built for a browser/internet experience half a generation removed from the present, and all the works of the Medalists.

Here’s Andrei Okounkov by Giovanni Felder, who first came to my attention because he was singled out by Ron Maimon in one of his answers on Quora. He’s a mathematical physicist of the highest order – well perhaps a notch or two below the Barry Simons of the world, but certainly the most promising under-40 representative of his subfield in 2006:

Andrei Okounkov’s initial area of research is group representation theory, with particular emphasis on combinatorial and asymptotic aspects. He used this subject as a starting point to obtain spectacular results in many different areas of mathematics and mathematical physics, from complex and real algebraic geometry to statistical mechanics, dynamical systems, probability theory and topological string theory.

The research of Okounkov has its roots in very basic notions such as partitions, which form a recurrent theme in his work. Partitions are a basic combinatorial notion at the heart of the representation theory. Okounkov started his career in this field in Moscow where he worked with G. Olshanski, through whom he came in contact with A. Vershik and his school in St. Petersburg, in particular S. Kerov.

The research programme of these mathematicians, to which Okounkov made substantial contributions, has at its core the idea that partitions and other notions of representation theory should be considered as random objects with respect to natural probability measures. This idea was further developed by Okounkov, who showed that, together with insights from geometry and ideas of high energy physics, it can be applied to the most diverse areas of mathematics.

Concluding remarks on Andrei:

Andrei Okounkov is a highly creative mathematician with both an exceptional breadth and a sense of unity of mathematics, allowing him to use and develop, with perfect ease, techniques and ideas from all branches of mathematics to reach his research objectives. His results not only settle important questions and open new avenues of research in several fields of mathematics, but they have the distinctive feature of mathematics of the very best quality: they give simple complete answers to important natural questions, they reveal hidden structures and new connections between mathematical objects and they involve new ideas and techniques with wide applicability.

Moreover, in addition to obtaining several results of this quality representing significant progress in different fields, Okounkov is able to create the ground, made of visions, intuitive ideas and techniques, where new mathematics appears. A striking example for this concerns the relation to physics: many important developments in mathematics of the last few decades have been inspired by high energy physics, whose intuition is based on notions often inaccessible to mathematics. Okounkov’s way of proceeding is to develop a mathematical intuition alternative to the intuition of high energy physics, allowing him and his collaborators to go beyond the mere verification of predictions of physicists. Thus, for example, in approaching the topological vertex of string theory, instead of stacks of D-branes and low energy effective actions we find mathematically more familiar notions such as localization and asymptotics of probability measures. As a consequence, the scope of Okounkov’s research programme goes beyond the context suggested by physics: for example the Maulik–Nekrasov– Okounkov–Pandharipande conjecture is formulated (and proved in many cases) in a setting which is much more general than the Calabi–Yau case arising in string theory.

Here’s Wendelin Werner, by Courant Institute director Charlie Newman, noting that this was the first time a probabilist had won the Fields (turns out they’re even more slighted than logicians huh?):

It is my great pleasure to briefly report on some of Wendelin Werner’s research accomplishments that have led to his being awarded a Fields Medal at this International Congress of Mathematicians of 2006. There are a number of aspects of Werner’s work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion – which, as we shall see, plays a major role in his later work as well. Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history.

I myself was originally trained, not in Probability Theory, but in Mathematical Physics. Werner’s work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Mapping Theory to fundamental issues in Statistical Physics, as we shall discuss. A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are providing quite new conceptual understanding of basic phenomena – in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation – see Figure 1.

Permit me a somewhat more personal remark as director of the Courant Institute for the past four years. We have a scientific viewpoint, as did our predecessor institute in Göttingen – namely, that an important goal should be the elimination of artificial distinctions between the Mathematical Sciences and their applications in other Sciences – I believe Wendelin Werner’s work brilliantly lives up to that philosophy.

Yet a third source of pleasure concerns the collaborative nature of much ofWerner’s work. Beautiful and productive mathematics can be the result of many different personal workstyles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work.

Here’s the star-who-didn’t-turn-up of the 2006 ICM show, the sensational Grigori Perelman, by John Lott. Lott’s exposition is pretty dry – I’m getting used to the personality-filled writings of the Gowers and Baezes and Mazurs and Vakils of the world, it seems:

Grigory Perelman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.

Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture.

In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture.

Perelman’s papers have been scrutinized in various seminars around the world. At the time of this writing, the work is still being examined.

Grigory Perelman has revolutionized the fields of geometry and topology. His work on Ricci flow is a spectacular achievement in geometric analysis. Perelman’s papers show profound originality and enormous technical skill. We will certainly be exploring Perelman’s ideas for many years to come.

And here, finally, is Charlie on Terry. Charlie’s writing has character. When I read him I can’t help but like him. He’s like Ravi Vakil in that regard, and like Scott Alexander from back in the day. His style is deceptively simple, in fact to a degree I’ve almost never seen. Maybe it’s the short sentences and simple words.

Mathematics at the highest level has several flavors. On seeing it, one might say:

(A) What amazing technical power!
(B) What a grand synthesis!
(C) How could anyone not have seen this before?
(D) Where on earth did this come from?

The work of Terence Tao encompasses all of the above. One cannot hope to capture its extraordinary range in a few pages. My goal here is simply to exhibit a few contributions by Tao and his collaborators, sufficient to produce all the reactions (A)… (D). I shall discuss the Kakeya problem, nonlinear Schrödinger equations and arithmetic progressions of primes.

I have repeatedly used the phrase “tour-de-force”; I promise that I am not exaggerating.

There are additional first-rate achievements by Tao that I have not mentioned at all. For instance, he has set forth a program [22] for proving the global existence and regularity of wave maps, by using the heat flow for harmonic maps. This has an excellent chance to work, and it may well have important applications in general relativity. I should also mention Tao’s joint work with Knutson [19] on the saturation conjecture in representation theory. It is most unusual for an analyst to solve an outstanding problem in algebra.

Tao seems to be getting stronger year by year. It is hard to imagine what can top the work he has already done, but we await Tao’s future contributions with eager anticipation.