Winners of the 2010 IMU Prizes

The IMU Awards

The following four awards instituted by the International Mathematical Union (IMU) were given away at the 2010 International Congress of Mathematicians (ICM 2010) being held during August 19-27 at Hyderabad, India.

I. Fields Medals

II. The Rolf Nevanlinna Prize

III. The Gauss Prize

IV. The Chern Medal Award

The ICM takes place every four years. By tradition, the awards are given away during the opening ceremony of the ICM. At ICM 2010, the Honourable President of India, Smt. Pratibha Patil, presented the awards during the inauguration of the Congress on August 19.

(For more information about the goals, rules and history of the prizes, see IMU home page: http://www.mathunion.org/general/prizes)

I. The Fields Medals

The Fields Medals, first awarded in 1936, is the most traditional and important international prize in the world of mathematics. Up to four medals are awarded every four years on the occasion of the quadrennial International Congress of mathematicians (ICM) to recognize outstanding mathematical achievements for existing work and for the promise of future achievement. To be eligible for the award, the candidate’s 40^{th} birthday must not have occurred before January 1, 2010.

The Fields Medals were first proposed at the 1924 ICM in Toronto, where it was resolved that, at each subsequent Congress, two gold medals should be awarded for outstanding mathematical achievement. The Canadian mathematician J.C. Fields, who was the Secretary of the 1924 Congress, later donated funds towards these awards. Fields considered two fundamental principles for the award: the solution of a difficult problem and the creation of a new theory enlarging the fields of application of mathematics. In 1966, in view of greatly expanding mathematical research, it was decided that up to four medals could be given.

The Fields Medal is made of gold and the award includes a cash prize of Canadian $15,000.

(http://www.mathunion.org/general/prizes)

The 2010 Fields Medals

The Four Fields Medal awardees chosen for the year 2010 are:

1. Elon Lindenstrauss of Hebrew University, Jerusalem, Israel;

2. Ngô B?o Châu of Université Paris-Sud, Orsay, France;

3. Stanislav Smirnov of Université de Genève, Switzerland; and,

4. Cédric Villani of Institut Henri Poincaré, Paris, France.

I.1 Elon Lindenstrauss

Lindenstrauss is being awarded the 2010 Fields Medal *“for his results on measure rigidity in ergodic theory, and their applications to number theory.”*

Lindenstrauss has made far-reaching advances in ergodic theory, the study of measure preserving transformations. His work on a conjecture of Furstenberg and Margulis concerning the measure rigidity of higher rank diagonal actions in homogeneous spaces has led to striking applications. Specifically, jointly with Einsiedler and Katok, he established the conjecture under a further hypothesis of positive entropy. It has impressive applications to the classical Littlewood

Conjecture in the theory of diophantine approximation. Developing these as well other powerful ergodic theoretic and arithmetical ideas, Lindenstrauss resolved the arithmetic quantum unique ergodicity conjecture of Rudnick and Sarnak in the theory of modular forms. He and his collaborators have found many other unexpected applications of these ergodic theoretic techniques in problems in classical number theory. His work is exceptionally deep and its impact goes far beyond ergodic theory.

Brief Bio:

Born in Jerusalem, 1970. Married, three children.

Education: B.Sc. in Mathematics and Physics, The Hebrew University of Jerusalem, Israel, 1991; M.Sc. in Mathematics, The Hebrew University, 1995; Ph.D. in Mathematics, The Hebrew University, 1999. Thesis Title: Entropy Properties of Dynamical Systems. Thesis advisor: Prof. Benjamin Weiss.

Academic Positions: Joint appointment as Professor, Hebrew University of Jerusalem, (October 1, 2008- ) and Professor, Princeton University, NJ (2004 – August 31, 2010). Visiting Member, Courant Institute, NYU (2003-2005). Clay Mathematics Institute Long Term Prize Fellow (2003-2005). Szego Assistant Professor, Stanford University, CA (2001-2003). Member, Institute for Advanced Study, Princeton NJ (1999-2001). Clore Fellowship, Hebrew University, Jerusalem (1998-1999). Teaching Assistant, Hebrew University, Jerusalem (1997-1998).

Scholarships and Awards: The Anna and Lajos Erds Prize in Mathematics 2009; Michael Bruno Memorial Award (given by the Rothschild “Yad Hanadiv” Foundation) 2008; European Mathematical Society Prize 2004; Salem Prize 2003; Clay Mathematical Institute Long Term Prize Fellow 2003-2005; Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics 2001; Nessyahu Prize for Ph.D. Thesis 2001; Kennedy-Lee Prize for Ph.D. thesis 2000; Charles Clore Scholarship (three years award) 1998;

Yashinski prize for excellence in graduate studies 1998.

I.2 Ngô B?o Châu

Ngô is being awarded the 2010 Fields Medal for *“for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.”*

In the 1960’s and 70’s Robert Langlands formulated various basic unifying principles and conjectures relating automorphic forms on different groups, Galois representations and L-functions. These led to what today is referred to as the Langlands programme. The main tool in establishing some cases of these conjectures is the trace formula and in applying it for the above purposes a central difficulty intervenes: to establish some natural identities in harmonic analysis on local groups as well as ones connected to arithmetic geometric objects. This problem became known as the Fundamental Lemma. After many advances by a number of researchers in 2004, Laumon and Ngô established the Fundamental Lemma for a special family of groups, and recently Ngô established the Lemma in general.

Ngô’s brilliant proof of this important long standing conjecture is based in part on the introduction of novel geometric objects and techniques into this sophisticated analysis. His achievement, which lies at the crossroads between algebraic geometry, group theory and automorphic forms, is leading to many striking advances in the Langlands programme as well as the subjects linked with it.

Brief Bio:

Ngô B?o Châu was born on June 28, 1972, in Hanoi, Vietnam. After secondary school in Vietnam, he moved to France and studied at the Université Paris 6, Ecole Normale Supérieure de Paris. He completed his PhD Degree in Orsay under the supervision of Gérard Laumon. He is currently Professor in the Faculté des Sciences at Orsay and Member of the Institute for Advanced Study in Princeton. In September 2010, he will start his new appointment at the University of Chicago. Jointly with Laumon, Ngô was awarded the Clay research award in 2004. In 2007, he was awarded the Sophie Germain prize and the Oberwolfach prize.

I.3 Stanislav Smirnov

Smirnov is being awarded the 2010 Fields medal *“for the proof of conformal invariance of percolation and the planar Ising model in statistical physics.”*

It was predicted in the 1990’s, and used in many studies, that the scaling

limit of various two dimensional models in statistical physics has an unexpected symmetry, namely it is conformally invariant. Smirnov was the first to prove this rigorously for two important cases, percolation on the triangular lattice and the planar Ising model. The proof is elegant and it is based on extremely insightful combinatorial arguments. Smirnov’s work gave the solid foundation for important methods in statistical physics like Cardy’s Formula, and provided an all-important missing step in the theory of Schramm-Loewner Evolution in the scaling limit of various processes.

Brief Bio:

Stanislav Smirnov was born in 1970 in St. Petersburg, Russia. He attended the 239^{th} mathematical school and St. Petersburg State University where he studied mathematical analysis with Viktor Havin. After graduating in 1992 he moved to the California Institute of Technology where he received his Ph. D. in 1996 under the direction of Nikolai Makarov. Smirnov spent an important part of his career in Stockholm, Sweden, where he came in 1998 after holding a Gibbs instructorship at Yale as well as short time positions at the Institute for Advanced Study, Princeton, and the Max Planck Institute for Mathematics (MPIM), Bonn. He became professor at the Royal Institute of Technology and researcher at the Swedish Royal Academy of Sciences in 2001. Since 2003 he is a professor at the University of Geneva, Switzerland.

Smirnov works in analysis, dynamical systems, probability and mathematical physics. His distinctions include the St. Petersburg Mathematical Society Prize (1997), the Clay Research Award (2001), the Salem Prize (2001), the Gran Gustafsson Research Prize (2001), the Rollo Davidson Prize (2002), the European Mathematical Society Prize (2004). His research is supported by the European Research Council and the Swiss National Science Foundation.

I.4 Cédric Villani

Villani is being awarded the 2010 Fields Medal *“for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.”*

One of the fundamental and initially very controversial theories of classical physics is Boltzmann’s kinetic theory of gases. Instead of tracking the individual motion of billions of individual atoms it studies the evolution of the probability that a particle occupies a certain position and has a certain velocity. The equilibrium probability distributions are well known for more than a hundred years, but to understand whether and how fast convergence to equilibrium occurs has been very difficult. Villani (in collaboration with Desvillettes) obtained the first result on the convergence rate for initial data not close to equilibrium. Later in joint work with his student Mouhut he rigorously established the so-called non-linear Landau damping for the kinetic equations of plasma physics, settling a long-standing debate. He has been one of the pioneers in the applications of optimal transport theory to geometric and functional inequalities. He wrote a very timely and accurate book on mass transport.

Brief Bio:

Cédric Villani was born in 1973 in France. After studying mathematics

at École Normale Supérieure in Paris from 1992 to 1996, he was appointed assistant professor there. He received his PhD in 1998. Since 2000 he has been a full professor at École Normale Supérieure de Lyon. He held semester-long visiting positions in Atlanta (1999), Berkeley (2004) and Princeton (2009), wrote about 50 research papers, and two reference books on optimal transport theory. His awards include the Jacques Herbrand Prize of the French Academy of Science (2007), the Prize of the European Mathematical Society (2008), the Henri Poincaré Prize of the International Association for Mathematical Physics, and the Fermat Prize (2009). In 2009 he was appointed director of the Institut Henri Poincaré (IHP) in Paris, and part-time visitor of the Institut des Hautes Études Scientifiques (IHES).

II. The Rolf Nevanlinna Prize

The Rolf Nevanlinna Prize, first awarded in 1982, recognizes outstanding contributions in Mathematical Aspects of Information Sciences. This includes complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence, scientific computing and numerical analysis, computer algebra and computational aspects of optimization and control theory.

The Nevanlinna Prize is awarded once every four years at the International Congress of Mathematicians (ICM). To be eligible, the candidate’s 40^{th} birthday must not have occurred before January 1, 2010. The Prize consists of a gold medal and a cash prize of € 10,000. Established in 1981, the Prize is named after Rolf Nevanlinna (1895-1980), one of the most famous Finnish mathematicians, who had been Rector of Helsinki University and President of the International Mathematical Union (IMU) during 1959-62. The Prize is financed by the University of Helsinki.

(http://www.mathunion.org/general/prizes)

The 2010 Rolf Nevanlinna Prize

Daniel Spielman of Yale University, USA, has been chosen for the 2010 Rolf Nevanlinna Prize *“for smoothed analysis of Linear Programming, algorithms for graph-based codes and applications of graph theory to Numerical Computing.”*

Linear Programming is one of the most useful tools in applied mathematics. The oldest algorithm for Linear Programming, the Simplex Method, works very well in practice, but mathematicians have been perplexed about this efficacy and have tried for long to establish this as a mathematical theorem. Spielman and his co-author Shenhua Teng developed a beautiful method and proved that, while there may be pathological examples where the method fails, slight modifications of any pathological example yields a “smooth” problem on which the Simplex method works very well.

A second major contribution of Spielman is in the area of coding. Much of the present-day communication uses coding, either for preserving secrecy or for ensuring error correction. An important technique to make both coding and decoding efficient is based on extremely well-connected graphs called expanders. Spielman and his co-authors have done foundational work on such codes and have designed very efficient methods for coding and decoding. These codes provide an efficient solution to problems such as packet-loss over the internet and are particularly useful in multicast communications. They also provide one of the best known coding techniques for minimizing power consumption required to achieve reliable communication in the presence of white Gaussian noise.

Brief Bio:

Daniel Alan Spielman was born in Philadelphia in March 1970. He received his B.A. in mathematics and Computer Science from Yale in 1992, and his Ph. D. in Applied Mathematics from M. I. T. in 1995. His thesis was on ‘Computationally Efficient Error-correcting Codes and Holographic Proofs’. He spent a year as a National Science Foundation (NSF) post-doctoral fellow in the Computer Science Department at University of California, Berkeley, and then taught at the Applied Mathematics Department of MIT until 2005. Since 2006 he has a Professor of Applied Mathematics and Computer Science at Yale University.

Spielman has won several awards and honours earlier, chief among them being the Fulkerson Prize, jointly awarded by the American Mathematical Society (AMS) and the Mathematical Programming Society (MPS) in 2009; the Gödel Prize, jointly awarded by the Association for Computing Machinery (ACM) and the European Association for Theoretical Computer Science (EATCS) for Spielman and Teng’s paper on ‘Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time’.

Spielman has applied for five patents for error-correcting codes that he has invented and four of them have already been granted by the U. S. Patent Office.

III. The Carl Friedrich Gauss Prize for Applied Mathematics

The Carl Friedrich Gauss Prize for Applied Mathematics was awarded for the first time at the last International Congress of Mathematicians in 2006 (ICM 2006) at Madrid, Spain. It has been created to emphasize that mathematics is a driving force behind many modern technologies. The Prize honours scientists whose life-long mathematical research has had an impact outside mathematics – in technology, in business or simply in people’s every day lives.

The Gauss Prize is awarded jointly by the Deutsche Mathematiker-Vereinigung (DMV) – the German Mathematical Union — and the International Mathematical Union (IMU) and is administered by the DMV.

The Award is named after Carl Friedrich Gauss (1777-1855) who was one of the greatest mathematicians of all time. He combined scientific theory and practice like no other before him, or since. His *Disquisitiones Arithmeticae*, published in 1801, stands to this day as a true masterpiece of scientific investigation. Gauss’s name is associated with the Method of Least Squares widely used in data fitting, the bell-shaped probability distribution called the Gaussian, the electric telegraph and the unit of magnetic induction Gauss in electromagnetism.

The Gauss Prize consists of a gold medal and a cash award of € 10,000. The source of the prize is a surplus from the International Congress of Mathematicians (ICM 1998) held in Berlin.

There is no age limit for the Gauss Prize.

(http://www.mathunion.org/general/prizes)

The 2010 Gauss Prize

Yves Meyer, Professor Emeritus at École Normale Supérieure de Cachan, France, has been selected for the 2010 Gauss Prize “*for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis”*.

Meyer has made fundamental contribution to a number of mathematical areas. Around 1970, he developed the theory of model sets in number theory, which has become an important tool in the mathematical study of quasicrystals — space-filling structures that are ordered but lack translational symmetry — and aperiodic order in general. Together with Ronald Coifman and Alan MacIntosh he proved the continuity of the Cauchy integral operator on all Lipschitz curves, a long-standing problem in analysis.

Meyer played a leading role in the modern development of wavelet theory, which has had a spectacular impact in information sciences, statistics and technology. Fourier analysis is a universal tool in applied mathematics, and due in a large measure to Meyer’s work, wavelet theory has become the new name for Fourier analysis. He constructed the first non-trivial wavelet bases and wavepackets that dramatically extended the expressing power of wavelets. This led to many applications in practice – in image processing, data compression, statistical data analysis and elsewhere. Among the many applications of Meyer’s work, the techniques for restoring satellite images and the image compression standard JPEG-2000 deserve particular mention.

More recently, he has found a surprising connection between his early work on the model sets used to construct quasicrystals — the ‘Meyer Sets’ — and ‘compressed sensing’, a technique used for acquiring and reconstructing a signal utilizing the prior knowledge that it is sparse or compressible. Based on this he has developed a new algorithm for image processing. A version of such an algorithm has been installed in the space mission Herschel of the European Space Agency (ESA), which is aimed at providing images of the oldest and coldest stars in the universe.

Brief Bio:

Yves Meyer was born on July 19, 1939. He graduated from École Normale Supérieure, Paris, in 1960 and became a high school teacher until 1963. He then obtained a teaching assistantship at Université de Strasbourg from where he obtained his Ph. D. in 1966. He was his own thesis supervisor, which, according to him, was not uncommon those days. He has been a professor at École Polytechnique, Université Paris-Dauphine and has also held a full research position at *Centre National de la Recherche Scientifique* (CNRS). His current position as Professor Emeritus at École Normale Supérieure de Cachan, France, comes after having served as a professor during 1999-2009 at the same institution. He is a foreign honorary member of the American Academy of Arts and Sciences. He has also been awarded a Doctorate (*Honoris causa*) by Universidad Autonoma de Madrid.

IV. The Chern Medal Award

The Chern Medal Award, established in the memory of the outstanding Chinese mathematician Shiing-Shen Chern (1911 – 2004), will be awarded for the first time at the 2010 International Congress of Mathematicians (ICM 2010) to be held during August 19-27 at Hyderabad, India. It will subsequently be awarded every four years at the quadrennial ICMs.

The Award will be given to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics. All persons, regardless of age or vocation, are eligible for the Medal.

The award consists of a gold medal and a cash prize of US$ 250,000. In addition, each Medalist may nominate one or more organizations to receive funding totaling US$ 250,000, for the support of research, education, or other outreach programmes in the field of mathematics. This is called the “Organization Award”. The prize is jointly awarded by the International Mathematical Union (IMU) and the Chern Medal Foundation (CMF). The CMF funds the award.

At ICM 2010 there will also be a separate ceremony to commemorate S. S. Chern, who devoted his life to mathematics, both in active research and education, and in nurturing the field whenever the opportunity arose. Chern obtained fundamental results in all the major aspects of modern geometry, and founded the area of global differential geometry. His work exhibited keen aesthetic tastes in his selection of problems and in his breadth exemplified the interconnectivity of modern geometry and all of its aspects.

(http://www.mathunion.org/general/prizes)

The 2010 Chern Medal Award

Louis Nirenberg of Courant Institute of Mathematical Sciences, New York University (NYU), has been selected to be the first recipient of the Chern Medal *“for his role in the formulation of the modern theory of non-linear elliptic partial differential equations and for mentoring numerous students and post-docs in this area”*.

Nirenberg is one of the outstanding analysts and geometers of the 20^{th} Century and his work has had a major influence in the development of several areas of mathematics and their applications. He has made fundamental contributions to the understanding of linear and non-linear partial differential equations (PDEs) and related aspects of complex analysis and geometry, the basic mathematical tools of modern science. He developed intricate connections between analysis and differential geometry and applied them to the theory of fluid flow and other physical phenomena.

Nirenberg’s name is associated with several major developments in analysis in the last 65 years. His theorem with August Newlander on the existence of almost complex structures has become a classic. One of the most widely quoted results in analysis is that *a priori *estimates for general linear elliptic systems, which he obtained with Shmuel Agmon and Avron Douglis. His fundamental work with Fritz John on functions of bounded mean oscillation was crucial for later work of Charles Fefferman on the space of such functions. In collaboration with Joseph Kohn, he introduced the nation of pseudo-differential operator, which has been influential in many areas of mathematics. Other significant works of Nirenberg, which he has carried out in collaboration with others, have been on solvability of PDEs, existence of smooth solutions of a class of PDEs and equations of fluid motion of Navier-Stokes kind.

Nirenberg is recognized for his excellent lectures and lucid expository writing. He has published over 185 papers and has had 46 students and 245 descendents according to the Mathematics Genealogy Project. In each of the last 10 years, top 15 cited papers in mathematics include at least 2 of Nirenberg, according to MathSciNet.

Kohn has said this of him: “Nirenberg’s career has been an inspiration; his numerous students, collaborators, and colleagues have learned a great deal from him. Aside from mathematics, Nirenberg has taught all of us the enjoyment of travel, movies, and gastronomy. An appreciation of Nirenberg also must include his ever-present sense of humour. His humour is irrepressible, so that on occasion it makes its way to the printed page.”

In an interview in April 2002, Nirenberg said: “I wrote one paper with Philip Hartman that was elementary but enormous fun to do. That’s the thing I try to get across to people who don’t know anything about mathematics, what fun it is! One of the wonders of mathematics is you go somewhere in the world and you meet other mathematicians, and it’s like one big family. This large family is a wonderful joy.”

Brief Bio:

Nirenberg was born on February 25, 1925, in Hamilton, Ontario, Canada. After receiving his bachelor’s degree from McGill University in 1945, he went to NYU from where he obtained his M. S. in 1947 and Ph. D in 1949, under the direction of James Stoker. Nirenberg then joined the faculty of NYU. He was one of the original members of the Courant Institute of Mathematical Sciences. After spending his entire academic career at Courant, he retired in 1999. He is now an Emeritus Professor there.

Nirenberg has received several awards and honours, chief among them being the American Mathematical Society’s Bôcher Prize in 1959 for his work on PDEs, Jeffrey-Williams Prize of the Canadian Mathematical Society in 1987 and the Steele Prize of the AMS in 1994 for Lifetime Achievement. In 1982 he (along with late V. I. Arnold) was the first recipient in mathematics of the Crafoord Prize, established by the Royal Swedish Academy of Sciences in areas not covered by the Nobel Prizes. In 1995 he received the National Medal of Science, the highest honour in the U. S. for contributions to science.