1.1. Alumni Prize (awarded by the Belgium Scientific Research Fund), 1979.

1.2. Empain Prize (awarded by the Belgium Scientific Research Fund), 1983.

1.3. Salem Prize (awarded by the School of Mathematics at the Institute for Advanced Study in Princeton), 1983.

1.4. The Dr A De Leeuw-Damry-Bourlart Excellence Prize (awarded by the Belgian Scientific Research Fund), 1985 (quinquennial Belgian Science Prize).

1.5. Langevin Prize (awarded by the French Académie des sciences), 1985.

1.6. Elie Cartan Prize (awarded by the French Académie des sciences), 1990 (triennial).

1.7. Ostrowski Prize (awarded by the Ostrowski Foundation, Basel-Switzerland), 1991 (biennial).

1.8. Fields Medal (awarded by the International Mathematical Union) ICM Zurich, 1994.

1.9. V I Vernadski Gold Medal (awarded by the National Academy of Sciences of Ukraine), 2009.

1.10. Shaw Prize in Mathematical Sciences (awarded by The Shaw Prize Foundation), 2010.

1.11. Crafoord Prize (awarded by the Royal Swedish Academy of Sciences), 2012.

1.12. Title of Baron granted by King of Belgium, July 2015.

1.13. Antonio Feltrinelli International Prize for Mathematics (awarded by the Accademia Nazionale dei Lincei), 2016.

1.14. Breakthrough Prize in Mathematics (awarded by the Breakthrough Prize Board), 2017.

1.15. Leroy P Steele Prize for Lifetime Achievement (awarded by the American Mathematical Society), 2018.

2.1. Contribution of Jean Bourgain for the Shaw Prize 2010.

Mathematical analysis deals with limiting processes such as the approximation of a circle by inscribed regular polygons with increasing numbers of sides (a method used by Archimedes), or the notion of instantaneous velocity used in dynamics. The calculus of Newton and Leibniz provided the machinery for its successful application, from the orbits of planets, to flight of aeroplanes and the devastation of a tsunami.

Underpinning this limiting process is a variety of inequalities, often of a combinatorial nature, whose precise formulation and proof require great insight and ingenuity. The tools and language of analysis form the foundation for vast areas of mathematics, ranging from probability theory and statistical physics to partial differential equations, dynamical systems, combinatorics and number theory.

Jean Bourgain is one of the most brilliant analysts of our times. He has resolved central and long-standing problems in each of the above fields. In doing so he has introduced fundamental techniques many of which have become standard tools in these areas. His work and ideas have greatly enhanced the very fruitful cross fertilisations between all these disciplines.

A prime example of his work is his development of the sum product phenomenon. This is a fundamental combinatorial property which quantifies the relation between the two most basic operations of addition and multiplication. He has used this sum-product theory to resolve problems connected with distribution and counting of symmetries, combinatorics, number theory and solutions of algebraic equations.

More surprisingly, these techniques of Bourgain are intimately related to the very subtle geometry of the Kakeya problem, where a car (idealised as a line segment) is to be reversed in an arbitrarily small area, using an $N$-point turn with very large $N$.

In many areas of mathematics and science, random numbers play a key role, but they are in fact hard to produce: tossing a coin is not a practical solution and the coin may be biased. Bourgain has applied his techniques to provide explicit structures that exhibit randomness, and these have important applications in theoretical computer science.

Mathematical Sciences Selection Committee
The Shaw Prize

27 May 2010, Hong Kong

2.2. An Essay on the Shaw Prize 2010 for Jean Bourgain.

Mathematical analysis is concerned with the study of infinite processes, and the differential calculus of Newton and Leibniz lies at its heart. It provided the foundation and the language for Newtonian mechanics and the whole of mathematical physics. Over the past three centuries it has permeated much of mathematics and science.

Associated with the limiting process there are many technically difficult "estimates" or inequalities, of a combinatorial or algebraic nature, which prepare the ground and justify passing to the limit. Such estimates are often extremely hard since they address some subtle and important aspect of the problem at hand. Establishing this becomes a key step, opening the door to a wide variety of applications.

Over the past thirty years this study has undergone a mini-revolution in which a succession of hard problems of this nature have been solved, using a variety of novel techniques and ideas which often cross disciplinary boundaries and stimulate cross-fertilisation.

Jean Bourgain is one of the leading analysts in the world today and he has played a major role in this revolution. He is much admired especially by those who make regular use of the multitude of powerful techniques that he has provided. He has written over 350 papers, each of which is first rate, and a number of which contain solutions of central long-standing problems.

The fields in which he has made such fundamental contributions include harmonic analysis, functional analysis, ergodic theory, partial differential equations, mathematical physics, combinatorics and theoretical computer science. Some of the well-known problems that he has solved include the embedding, with least distortion, of finite metric spaces in Hilbert space; extending the validity of Birkhoff's ergodic theorem to very general sparse arithmetic sequences; and the boundedness in $L^{p}$ of the circular maximum function in two dimensions.

He has also made a fundamental breakthrough in the study of the non-linear Schrödinger equation for the critical exponent defocusing case, introducing new tools which have led to significant progress on this difficult problem.

A whole area where Bourgain has led the way, and which deserves special mention, is the field of arithmetic combinatorics and its applications. A notable example is his solution of the "local" version of the Erdös-Volkmann conjecture. The original conjecture asserts that any measurable subring of the real line has dimension either 0 or 1. This was proved by Edgar and Miller in 2003 and, around the same time, Bourgain established the local version. This provides a sharp and powerful quantification of this phenomenon and is technically a tour de force.

In 2004 Bourgain, Katz and Tao proved their celebrated finite field analogue, known as the "Sum Product theorem". This is an elementary and fundamental quantification of the fact that finite fields have no subrings, and it measures a basic disjunction between the operations of addition and multiplication in a finite field. Bourgain has developed and extended this phenomenon making it into a theory. A first application is to estimating algebro-geometric character sums. For this the standard tool has been the famous solution of the Weil conjectures, established by Deligne using Grothendieck's cohomology theory. However for these methods to give non-trivial information one needs the Betti numbers of the corresponding varieties to be small compared to the size of the finite field. What is remarkable about Bourgain's results is that they give results even when the Betti numbers are big.

Another application of Bourgain's theory, developed in collaboration with Gamburd is a proof of the expander conjecture of Lubotsky for the group $SL_{2}(F_{p})$ and the spectral gap conjecture for elements in the group $SU(2)$. These are concerned with the spectra of the images, in high dimensional representations of these groups, of elements of their group rings. They yield exponentially sharp equidistribution rates for random walks in these groups and are central to problems such as classical sieving in number theory and to aperiodic tilings of 3-dimensional space.

Bourgain has also developed some striking applications of his theory to theoretical computer science by giving a much sought after explicit construction of pseudorandom objects called extractors. These, as well as the expanders, are basic building blocks used in fast derandomization algorithms.

Bourgain's spectacular contributions to modern mathematics make him a very deserving winner of the 2010 Shaw Prize in the Mathematical Sciences.
Mathematical Sciences Selection Committee

The Shaw Prize
28 September 2010, Hong Kong

2.3. Autobiography of Jean Bourgain for the Shaw Prize 2010.

I was born in 1954 in Oostende (Belgium) from a family of medical doctors. My mother was a paediatrician and father a professor of physiology. This is where I completed elementary and high school until enrolling in 1971 at the Free University of Brussels as a student in mathematics. My interests in mathematics had started a few years earlier, perhaps mostly from browsing through calculus books we had at home.

Classes at the Free University were relatively small which allowed for more individualised attention than what happens at most larger institutions. I obtained the degree of 'Licentiaat' in 1975 and started working in areas such as descriptive set theory and functional analysis. In 1977 I got my PhD degree and a 'Habilitation' in 1979 for work on the structural theory of Banach Spaces and the relation between their local and infinite dimensional properties.

From 1975 on (until 1984) I was fortunate to have a position at the Belgian science foundation, which allowed me to do research and travel without other duties. Frequent visits to French institutions (such as the Centre de mathématiques of the École Polytechnique) and Israel (Jerusalem and Tel Aviv) enlarged my professional contacts and interests. The year 1984-85 was particularly important for me. I was a visiting member at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, participating in a special year on high dimensional convexity. In joint work with V Milman, using these methods we resolved an old problem of K Mahler on the volume of convex bodies and their polar, proving a converse to Santalo's inequality. One of the original motivations lies in the geometry of numbers, but our work became later also important to theoretical computer science.

In 1985, I was appointed to the IHÉS faculty and the same year also started a half time position at the University of Illinois as J L Doob Professor. My research interests had evolved towards harmonic analysis, ergodic theory and partial differential equations; in France, there were active groups in Orsay, the IHP institute and the University of Paris 7. Besides my IHÉS colleagues, I enjoyed frequent discussions with people like M Herman and J-P Thouvenot.

Starting from the early nineties, I have spent a great deal of time working on various aspects of Hamiltonian evolution equations. Putting aside the integrable cases, which are special, the available conserved quantities may not suffice to establish solutions or can be inadequate to deal with important classes of initial data. They also do not shed much light on how the solutions behave for large time. By bringing into play methods from probability and smooth dynamical systems, further information can sometimes be obtained. In this context, I succeeded using my work on 'Fourier restriction phenomena' to establish a well-defined dynamics on the support of the Gibbs-measure, which plays the role of an invariant measure for the flow. Some special cases had been studied earlier by J Lebowitz and his collaborators and qualitative results in this direction had also been obtained by P Malliavin. But understanding the full extent to which Gibbs measures are a substitute for a conserved quantity, from a more classical perspective, was a challenge.

1994 is the year I joined the Institute for Advanced Study in Princeton as part of the faculty of the School of Mathematics. Scientific life there was (and is) particularly intense for me, due to the many seminars at the institute and also Princeton University and the exposure to an infinite stream of visiting members. Initially, I continued working on differential equations and problems in mathematical physics, developing a theory of quasi-periodic solutions for evolution equations in higher dimension and contributing to the spectral theory of lattice Schrödinger operators modelling transport in inhomogenous media.

A few years after my arrival at IAS, we started a new direction at the institute, which is theoretical computer science. It became part of the maths school. At the start, no one could predict how exactly the interaction with the core mathematical activities would evolve in the longer run. In my view, it has been amazing. And I greatly benefited from it. One sample of this has to do with my earlier work in harmonic analysis of Euclidean space and the so-called 'Kakeya-set' problem, which plays a crucial role. Understanding the structure of 3-dimensional Kakeya sets (these are simply sets containing a line segment in every direction) turned out to have an unexpected realm of connections that have occupied me over the last decade. Among them are developments in the combinatorial aspects of finite fields (the so-called 'sum-product' phenomena), the theory of exponential sums, pseudo-randomness in computer science and the expansion properties of Cayley graphs of linear groups.

Collaboration and discussions with some of my colleagues in and outside the institute have been very important to me and I am very grateful to them.

28 September 2010, Hong Kong

3.1. The Crafoord Prize 2012 awarded to Jean Bourgain.

The Royal Swedish Academy of Sciences has decided to award the Crafoord Prize in Mathematics 2012 to Jean Bourgain, Institute for Advanced Study, Princeton, USA and Terence Tao, University of California, Los Angeles, USA, "... for their brilliant and groundbreaking work in harmonic analysis, partial differential equations, ergodic theory, number theory, combinatorics, functional analysis and theoretical computer science."

3.2. The masters of mathematics, Jean Bourgain and Terence Tao.

This year's Crafoord Prize Laureates have solved an impressive number of important problems in mathematics. Their deep mathematical erudition and exceptional problem-solving ability have enabled them to discover many new and fruitful connections and to make fundamental contributions to current research in several branches of mathematics.

On their own and jointly with others, Jean Bourgain and Terence Tao have made important contributions to many fields of mathematics - from number theory to the theory of non-linear waves. The majority of their most fundamental results are in the field of mathematical analysis. They have developed and used the toolbox of analysis in ground-breaking and surprising ways. Their ability to change perspective and view problems from new angles has led to many remarkable insights, attracting a great deal of attention among researchers worldwide.

4.1. Jean Bourgain Awarded Breakthrough Prize in Mathematics 2017.

Jean Bourgain, IBM von Neumann Professor in the School of Mathematics, has received the 2017 Breakthrough Prize in Mathematics, for "multiple transformative contributions to analysis, combinatorics, partial differential equations, high-dimensional geometry, and number theory." A member of the Institute Faculty since 1994, Bourgain was cited by the prize committee for solving major problems in vector spaces, as well as expander graphs; proving one of the two most fundamental theorems in ergodic theory; and developing novel techniques and insights applicable across a wide range of fields. A 1994 recipient of the Fields Medal, Bourgain has produced important work with impact across theoretical computer science, group expansion, spectral gaps, and the theory of exponential sums in number theory.

4.2. Jean Bourgain response to winning the 2017 Breakthrough Prize in Mathematics.

It is of course an immense honour for me to be awarded the Breakthrough Prize and also an occasion to thank all those who helped me along my career. Over the years I have been fortunate to interact with several truly exceptional individuals who introduced me to different subjects and from whom I learned a lot. Collaborations on all levels played an important part in my work and are greatly valued. Appointments at research institutions such as the Institut des Hautes Études Scientifiques in Bures-sur-Yvette (France) and the Institute for Advanced Study in Princeton provided ideal conditions for a full dedication to mathematics. I am most grateful for their trust. Last but not least, thanks to my family for their love and continuous support over the years.

5.1. Jean Bourgain receives 2018 Steele Prize for Lifetime Achievement.

Jean Bourgain, IBM von Neumann Professor in the School of Mathematics at the Institute for Advanced Study, received the 2018 Steele Prize for Lifetime Achievement for the breadth of his contributions made in the advancement of mathematics.

Jean Bourgain was born in 1954 in Oostende, Belgium. He earned his PhD in 1977 under the supervision of Freddy Delbaen. From 1975 until 1984 he held a position at the Belgian Science Foundation. In 1985 he was appointed to the IHES faculty, and the same year he also started a half-time position at the University of Illinois as J L Doob Professor. He joined the Institute for Advanced Study in 1994 as part of the School of Mathematics.

Bourgain was elected Associé Entranger de l'Academie des Sciences in 2000, Foreign Member of the Polish Academy in 2000, Foreign Member of Academia Europea in 2008, Foreign Member of the Royal Swedish Academy of Sciences in 2009, Foreign Associate of the National Academy of Sciences in 2011, and Foreign Member of the Royal Flemish Academy of Arts and Sciences in 2013.

Bourgain has been awarded numerous prizes and awards, including the Alumni Prize, Belgium NSF (1979); the Empain Prize, Belgium NSF (1983); the Salem Prize (1983); the Leeuw-Damry-Bourlart Excellence Prize (1985); the Langevin Prize (1985); the Elie Cartan Prize (1990); the Ostrowski Prize (1991); the Fields Medal (1994); the V I Vernadski Gold Medal (2010); the Shaw Prize (2010); the Crafoord Prize (2012); the title of Baron of Belgium (2016); and the Breakthrough Prize in Mathematics (2017).

He is a giant in the field of mathematical analysis, which he has applied broadly and to great effect. In many instances, he provided foundations for entirely new areas of study and in other instances he gave mathematics new tools and techniques. He has solved long-standing problems in Banach space theory, harmonic analysis, partial differential equations, and Hamiltonian dynamics. Bourgain's work has had important consequences in probability theory, ergodic theory, combinatorics, number theory, computer science, and theoretical physics. His vision, technical power, and broad accomplishments are astounding.

5.2. Jean Bourgain's response to winning the 2018 Steele Prize.

I am deeply honoured and grateful to receive the 2018 Steele Prize for Lifetime Achievement. Over the years, I have been fortunate to meet and interact with some remarkable individuals, with different interests and styles, from whom I learned a lot. They played a decisive role in introducing me to new subjects and encouraging my research. A large part of my work is also the result of fruitful collaborations with both junior and senior researchers, sometimes over an extended period of time. I am most grateful to them.

Exceptional working conditions also allowed me full scientific dedication. At an early career stage, it was an appointment at the Belgian Science Foundation. Later, in the mid-1980s, a professorship at the IHES in Bures-sur-Yvette and at the University of Illinois Urbana-Champaign, and since 1994 at the Institute for Advanced Study in Princeton. The intensity of scientific life and exposure to new ideas they offer was and is a great experience, and I would like to thank them for their trust.

At the present time mathematics is an extremely active science and its future bodes well for its constant progress both for solving old problems and opening new areas of research.

5.3. About the Steele Prize for Lifetime Achievement.

Presented annually, the AMS Steele Prize for Lifetime Achievement is one of the highest distinctions in mathematics, and is awarded for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through PhD students. The prize will be awarded Thursday, January 11, 2018, at the Joint Mathematics Meetings in San Diego.