Jean Bourgain died on December 22, at the age of 64, after a long battle against cancer. Born in Ostend, Belgium, on February 28, 1954, he was recognized as one of the best mathematicians in the world. However, he was not a child prodigy, as much of his eventual competitors. It took more than normal to speak and did not excel in mathematics in his childhood. However, he came to occupy one of the few chairs of the Institute for Advanced Study in Princeton, United States, from 1994, the same year he won the Fields Medal.
Although there have been other scientists of the caliber of Bourgain throughout history, one of the things that distinguished him was his style of work. In past centuries, the best mathematicians, such as Euler or Gauss, were dedicated to identifying new problems and solving them, sometimes without the intervention of other contemporary researchers. With the passage of time, some of the problems that remained unresolved became challenges for future generations.
Nowadays, and despite the recent tendency to divide mathematics into pure and applied mathematics, there is another equally useful division (or useless, depending on how you look at it) that separates mathematics into construction of theories (theory building) and problem solving (problem solving). The first category is dedicated to the pioneer discovery, to the search of interesting new mathematics, while the second one tries to solve the already known problems. The truth is that most mathematicians do not dedicate themselves exclusively to one thing or the other (it is not easy to divide between pure and applied). For example, many times a researcher modifies an old problem before attacking it in a way that builds new theory while sheds light on the original problem. But you could say that Bourgain was the baron of the problem solvers – what to modify before attacking was not his thing.
One problem that interested Bourgain, among many others, was the restriction conjecture of Stein, and the related ones Kakeya guesses Y of Bochner-Riesz. All three have something to do with convergence of Fourier series and the three were resolved in two dimensions in the seventies of the last century; the first by Charles Fefferman, the second by Roy Davies Y Antonio Córdoba and the third by Lennart Carleson and Per Sjölin. However, and despite the many efforts of the community, the three remain unresolved in more dimensions.
Progress in this task can be measured with a number, the fractal dimension or the integrability exponent, depending on the problem. In each moment someone has the world record; that is, the best dimension of the dimension or exponent. Every ten years, approximately, Bourgain returned to the task and beat the records. These interventions used to coincide with the moments of defeat, when the community felt that it was impossible to advance further. Just that was the moment in which Bourgain reappeared with a new genius, breaking the framework in which everyone was thinking.
Getting an advance on an issue that many have already studied almost always requires something great; sometimes, even, the creation of a whole new theory. That's where the techniques and deeper implications usually appear, even if the problem itself does not have a priori applications. For example, the theory developed for the restriction conjecture ended up being fundamental in the resolution, given by Bourgain, Demeter and Guth, of an old Vinogradov problem of number theory. The recent desire of governments to prioritize applications (which encourages some purely fictitious) puts at risk this type of progress, which can lead to unexpected developments, but much more important, in the future.
A final curiosity of the culture of problem solving is that you work less in teams than in the rest of the scientific community. It is delicate to ask a PhD student to think of a problem that nobody has been able to solve for decades. In fact, Bourgain only had one student throughout his career. And although he attended international congresses, for example, he came to Spain to teach the colloquium José Luis Rubio Memorial of France, many mathematicians did not get to know him personally. Even so, they will miss their enormous influence.
Keith Rogers He is the scientific head of CSIC and member of ICMAT
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".
Editing and coordination: Agate Rudder (ICMAT)