### Jean Bourgain †

Born 28 February 1954 in Ostend (Belgium)

Died 22 December 2018

Fields Medal (1994) for his contributions to geometry of Banach spaces, convexity in high-dimensional spaces, harmonic analysis, ergodic theory and the theory of nonlinear evolution equations

Bourgain received his doctorate (1977) and habilitation (1979) at the Vrije University of Brussels, where he was professor of Mathematics from 1981 onwards. Between 1985 and 1995 he taught as a professor at both the University of Illinois at Urbana-Champaign and the Institute des Hautes Etudes Scientifiques in Bures-sur-Yvette in France. In 1988, Bourgain was Visiting Professor at the Hebrew University in Jerusalem; in 1991 he served as Fairchild Distinguished Professor at the California Institute of Technology. Since 1994, he has been a Permanent Fellow at the School of Mathematics at the Institute for Advanced Study at Princeton.

Bourgain has received numerous awards, including the Salem Price (1983), the Langevin Prize of the French Academy of Sciences (1985), the Damry-Deleeuw-Bourlart Price (1985), the Elie Cartan Prize of the Académie des Sciences (1990), the Ostrowski Prize (1991), the Shaw Prize (2010) and the Crafoord Prize (2012, together with Terence Tao). Bourgain is an associate member of the French Academy of Sciences (2000), foreign member of the Polish Academy of Sciences (2000) and the Royal Swedish Academy of Sciences (2009), Foreign Associate of the National Academy of Sciences (2011) and member of the Royal Flemish Academy of Belgium for Sciences and Arts (2012). The Hebrew University of Jerusalem (1991), the University of Marne-La-Vallée (1994), and the Vrije Universiteit Brussel (1995) have awarded him honorary doctorates.

In comparison with other colleagues Jean Bourgain has extremely wide-ranging interests and is exceptionally productive: he numbers about 400 publications over many mathematical areas. In particular, he has explored Banach spaces, worked in harmonic analysis, ergodic theory, spectral problems and nonlinear partial differential equations of mathematical physics, number theory and combinatorics.

A small example of his ideas, which now has far-reaching consequences in mathematics, is his sum-product theory in algebra. The idea is simple: one considers a finite field, ie a set of numbers which allows certain calculations (addition, multiplication) and which exhibits some further features. One then picks a subset A of numbers from that field to form two new sets – one containing all products of two numbers from A and one containing all sums of two numbers from A. In 2003 Jean Bourgain, together with Nets Katz and Terence Tao, proved the lower bound for the size of the larger of these two sets, depending on the size of the set A. This estimate was the starting point for a series of interesting results from combinatorics, algebra, and number theory; one can also estimate the number of symmetric solutions of algebraic equations using the sum-product theory. Bourgain even made connections to the Kakeya needle problem, the question of the shape of a curve within which a needle of a certain length can rotate freely. (This need not be a circle as one might think; in fact, the surface on which the needle can be turned can be made arbitrarily small.)

One of Bourgain’s other key results is his work on ergodic theory, which was recognised at the Fields Medal ceremony inter alia. Ergodic theory is popularly referred to as chaos theory, and Bourgain proved a generalisation of the Birkhoff ergodic theorem, one of the fundamental theorems of the theory. It says – in simple terms – that in any chaotic system, for each start point you will run through every trajectory with a finite certain probability. What is important is how one measures whether the point is on the trajectory. Bourgain generalised this measure in 1987 and proved his new ergodic theorem with a very complex proof using new, inventive methods.