The Abel Prize Laureates 2015

John Forbes Nash, Jr. Louis Nirenberg Princeton University, USA Courant Institute, New York University, USA www.abelprize.no John F. Nash, Jr. and Louis Nirenberg receive the Abel Prize for 2015 “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.” Nash’s work on realizing manifolds as large class of nonlinear elliptic equations will real algebraic varieties and the Newlander- exhibit the same symmetries as those that Citation Nirenberg theorem on complex structures are present in the equation itself. further illustrate the influence of both Far from being confined to the solutions laureates in geometry. of the problems for which they were Regularity issues are a daily concern devised, the results proved by Nash and in the study of partial differential equations, Nirenberg have become very useful tools sometimes for the sake of rigorous and have found tremendous applications The Norwegian Academy of Science and Isometric embedding theorems, proofs and sometimes for the precious in further contexts. Among the most Letters has decided to award the Abel Prize showing the possibility of realizing an qualitative insights that they provide about popular of these tools are the interpolation for 2015 to intrinsic geometry as a submanifold of the solutions. It was a breakthrough in inequalities due to Nirenberg, including Euclidean space, have motivated some of the field when Nash proved, in parallel the Gagliardo-Nirenberg inequalities and John F. Nash, Jr., Princeton University these developments. Nash’s embedding with De Giorgi, the first Hölder estimates the John-Nirenberg inequality. The latter and Louis Nirenberg, Courant Institute, theorems stand among the most original for solutions of linear elliptic equations in governs how far a function of bounded mean New York University results in geometric analysis of the twentieth general dimensions without any regularity oscillation may deviate from its average, century. By proving that any Riemannian assumption on the coefficients; among and expresses the unexpected duality of “for striking and seminal contributions geometry can be smoothly realized as a other consequences, this provided a the BMO space with the Hardy space H1. to the theory of nonlinear partial differential submanifold of Euclidean space, Nash’s solution to Hilbert’s 19th problem about the The Nash-De Giorgi-Moser regularity theory equations and its applications smooth (C∞) theorem establishes the analyticity of minimizers of analytic elliptic and the Nash inequality (first proven by to geometric analysis.” equivalence of Riemann’s intrinsic point integral functionals. A few years after Nash’s Stein) have become key tools in the study of view with the older extrinsic approach. proof, Nirenberg, together with Agmon and of probabilistic semigroups in all kinds of Partial differential equations are used to Nash’s non-smooth (C1) embedding Douglis, established several innovative settings, from Euclidean spaces to smooth describe the basic laws of phenomena theorem, improved by Kuiper, shows the regularity estimates for solutions of linear manifolds and metric spaces. The Nash- in physics, chemistry, biology, and other possibility of realizing embeddings that at elliptic equations with Lp data, which Moser inverse function theorem is a powerful sciences. They are also useful in the analysis first seem to be forbidden by geometric extend the classical Schauder theory and method for solving perturbative nonlinear of geometric objects, as demonstrated by invariants such as Gauss curvature; this are extremely useful in applications where partial differential equations of all kinds. numerous successes in the past decades. theorem is at the core of Gromov’s whole such integrability conditions on the data are Though the widespread impact of both Nash John Nash and Louis Nirenberg have theory of convex integration, and has also available. These works founded the modern and Nirenberg on the modern toolbox of played a leading role in the development of inspired recent spectacular advances theory of regularity, which has since grown nonlinear partial differential equations cannot this theory, by the solution of fundamental in the understanding of the regularity of immensely, with applications in analysis, be fully covered here, the Kohn-Nirenberg problems and the introduction of deep incompressible fluid flow. Nirenberg, with his geometry and probability, even in very theory of pseudo-differential operators must ideas. Their breakthroughs have developed fundamental embedding theorems for the rough, non-smooth situations. also be mentioned. into versatile and robust techniques, which sphere S2 in R3, having prescribed Gauss Symmetry properties also provide Besides being towering figures, have become essential tools for the study of curvature or Riemannian metric, solved the essential information about solutions as individuals, in the analysis of partial nonlinear partial differential equations. Their classical problems of Minkowski and Weyl of nonlinear differential equations, both differential equations, Nash and Nirenberg impact can be felt in all branches of the (the latter being also treated, simultaneously, for their qualitative study and for the influenced each other through their theory, from fundamental existence results by Pogorelov). These solutions were simplification of numerical computations. contributions and interactions. The to the qualitative study of solutions, both important, both because the problems were One of the most spectacular results in consequences of their fruitful dialogue, in smooth and non-smooth settings. Their representative of a developing area, and this area was achieved by Nirenberg in which they initiated in the 1950s at the results are also of interest for the numerical because the methods created were the right collaboration with Gidas and Ni: they Courant Institute of Mathematical Sciences, analysis of partial differential equations. ones for further applications. showed that each positive solution to a are felt more strongly today than ever before.

4 5 © Peter Badge/Typos 1 in coop. with the HLF – all rights reserved 2015 © NYU Photo Bureau: Hollenshead smoothness was ignored and the second in equations. Within a few months Nash had A biography a setting that maintained smoothness. proved the existence of these inequalities. In order to prove his second Unknown to him, the Italian mathematician embedding theorem, Nash needed to Ennio De Giorgi had already proved this, of John Forbes Nash, Jr. solve sets of partial differential equations using a different method, and the result is that hitherto had been considered known as the Nash-De Giorgi theorem. Alexander Bellos impossible to solve. He devised an Nash was not a specialist. He worked iterative technique, which was then on his own, and relished tackling famous modified by Jürgen Moser, and is now open problems, often coming up with John F. Nash, Jr. is one of a handful of Problem, which he wrote in his second known as the Nash–Moser theorem. The completely new ways of thinking. In 2002 mathematicians known outside academia, term as a graduate student at Princeton Abel Prize laureate Mikhail Gromov has Louis Nirenberg said: “About twenty years due to the 2001 film about him, A Beautiful University. This paper led to his interest said: “What [Nash] has done in geometry ago somebody asked me, ‘Were there any Mind, loosely based on Sylvia Nasar’s in the new field of game theory – the is, from my point of view, incomparably mathematicians you would consider as bestselling biography of the same name. mathematics of decision-making. Nash’s greater than what he has done in geniuses?’ I said, ‘I can think of one, and The Oscar-winning movie fictionalized Ph.D. thesis, Non-Cooperative Games, economics, by many orders of magnitude. that’s John Nash.’… He had a remarkable Nash’s path from brilliant Princeton student is one of the foundational texts of game It was an incredible change in attitude of mind. He thought about things differently to being awarded the 1994 Nobel Prize theory. It introduced the concept of an how you think about manifolds. You can from other people.” for economics. equilibrium for non-cooperative games, take them in your bare hands, and what In 1957 Nash married Alicia Larde, Inevitably, the Hollywood version of the “Nash equilibrium”, which has had a you do may be much more powerful than a physics major whom he met at MIT. In Nash’s life story differed from the real great impact in economics and the social what you can do by traditional means.” 1959 when Alicia was pregnant with their one in many ways. In particular, the film sciences. In the early 1950s Nash worked as a son, he began to suffer from delusions and focused on his early results in game theory, While at Princeton Nash also made his consultant for the RAND Corporation, a extreme paranoia and as a result resigned which have applications in economics, first breakthrough in pure mathematics. He civilian think-tank funded by the military in from the MIT faculty. For the next three and omitted his research into geometry described it as “a nice discovery relating Santa Monica, California. He spent a few decades Nash was only able to do serious and partial differential equations, which the to manifolds and real algebraic varieties.” summers there, where his work on game mathematical research in brief periods of mathematical community regards as his In essence the theorem shows that any theory found applications in United States’ lucidity. He improved gradually and by the most important and deepest work. manifold, a topological object like a surface, military and diplomatic strategy. 1990s his mental state had recovered. John Forbes Nash, Jr. was born in can be described by an algebraic variety, Nash won one of the first Sloan The 1990s also saw him receive a 1928 in Bluefield, West Virginia, a small, a geometric object defined by equations, Fellowships in 1956 and chose to take number of honours for his professional remote town in the Appalachians. His in a much more concise way than had a year’s sabbatical at the Institute of work. As well as winning the prize in father was an electrical engineer at the previously been thought possible. The Advanced Study in Princeton. He based economic sciences in memory of Alfred local power company and his mother a result was already regarded by his peers as himself not in Princeton, but in New York, Nobel in 1994, which he shared with John schoolteacher. He entered the Carnegie an important and remarkable work. where he spent much of his time at Richard C. Harsanyi and Reinhard Selten, he was Institute of Technology (now Carnegie In 1951 Nash left Princeton to take up Courant’s fledgling Institute for Applied elected a member of the National Academy Mellon University) in Pittsburgh with a full an instructorship at MIT. Here he became Mathematics at NYU. It was here Nash met of Sciences in 1996, and in 1999 he won scholarship, originally studying for a major interested in the Riemann embedding Louis Nirenberg, who suggested to him the American Mathematical Society’s Steele in chemical engineering, before switching problem, which asks whether it is possible that he work on a major open problem in Prize for Seminal Contribution to Research to chemistry and finally changing again to embed a manifold with specific rules nonlinear theory concerning inequalities for his 1956 embedding theorem, sharing to mathematics. about distance in some n-dimensional associated with elliptic partial differential it with Michael G. Crandall. At Carnegie, Nash took an elective Euclidean space such that these rules are course in economics, which gave him the maintained. Nash provided two theorems idea for his first paper, The Bargaining that proved it was true: the first when

8 9 although it only had a small number of staff. Ever since he spent the academic A biography Nirenberg was one of its leading lights, and year 1951–52 in Zürich, Switzerland, and the mathematician who did the most work Göttingen, Germany, Nirenberg has been in providing a theoretical grounding for a well-travelled and active member of the of Louis Nirenberg modern analysis of PDEs. international mathematical community. Nirenberg has always preferred to On his first professional visit to Italy, in Alexander Bellos work in collaboration, with more than 90 1954 to attend a conference on PDEs, he per cent of his papers written jointly (none, immediately felt surrounded by friends. however, with John F. Nash, Jr., whom “That’s the thing I try to get across to Louis Nirenberg has had one of the longest, Nirenberg went for an interview in New Nirenberg got to know well during the people who don’t know anything about most feted – and most sociable – careers York and was offered an assistantship. He academic year 1956–57). Important papers mathematics, what fun it is!” he has said. in mathematics. In more than half a century got his masters in 1947, and embarked include results with his student August “One of the wonders of mathematics is you of research he has transformed the field on a Ph.D. under James J. Stoker, who Newlander on complex structures in 1957, go somewhere in the world and you meet of partial differential equations, while his suggested to him an open problem in with Shmuel Agmon and Avron Douglis other mathematicians and it’s like one big generosity, gift for exposition and modest geometry that had been stated by Hermann on regularity theory for elliptic equations family. This large family is a wonderful joy.” charm have made him an inspirational Weyl three decades previously: can you in 1959, with Fritz John introducing the He was present at the first big US–Soviet figure to his many collaborators, students embed isometrically a two-dimensional function space of functions with bounded joint maths conference in Novosibirsk in and colleagues. sphere with positive curvature into three mean oscillation in 1961, with David 1963, and in the 1970s was one of the first Louis Nirenberg was born in Hamilton, Euclidean dimensions as a convex surface? Kinderlehrer and Joel Spruck developing US mathematicians to visit China. Canada, in 1925 and grew up in Montreal, In order to prove that you can, he reduced regularity theory for free boundary Nirenberg has gathered a significant where his father was a Hebrew teacher. the problem to one about nonlinear partial problems in 1978 and with Basilis Gidas number of prestigious accolades. He won His first interest in mathematics came differential equations. The PDEs in question and Wei Ming Ni about the symmetries the American Mathematical Society’s from his Hebrew tutor, who introduced were elliptic, a class of equations that have of solutions of PDEs in 1979. A paper on Bôcher Memorial Prize in 1959. In 1969 him to mathematical puzzles. He studied many applications in science. Nirenberg’s solutions to the Navier–Stokes equations, he was elected to the National Academy mathematics and physics at McGill subsequent work has been largely co-authored with Luis A. Caffarelli and of Sciences. He won the inaugural University, Montreal, avoiding the draft concerned with elliptic PDEs, and over Robert V. Kohn, won the American Crafoord Prize, awarded by the Royal during World War II thanks to Canada’s the following decades he developed many Mathematical Society’s 2014 Steele Prize Swedish Academy of Science and policy of exempting science students, and important theorems about them. for Seminal Contribution to Research. given in areas not covered by the Nobel graduated in 1945. Nirenberg never left mathematics, As well as demonstrating vision Prizes, in 1982 (together with Vladimir The summer after graduating Nirenberg nor indeed NYU. Once he got his Ph.D. in and leadership, Nirenberg has shown Arnold). He received the Steele Prize for worked at the National Research Council of 1949 he stayed on as a research assistant. remarkable energy and stamina, continuing Lifetime Achievement from the American Canada on atomic bomb research. One of He was a member of the faculty – known to produce ground-breaking work in Mathematical Society in 1994, and he the physicists there was Ernest Courant, the since 1965 as the Courant Institute of different areas of PDEs until his 70s. He is received the National Medal of Science elder son of New York University professor Mathematical Sciences – his entire career, known not only for his technical mastery in 1995, the highest honour in the US Richard Courant, who was building up becoming a full professor in 1957. Between but also for his taste, instinctively knowing for contributions to science. In 2010 he NYU’s mathematics department. Nirenberg 1970 and 1972 he was the Institute’s which are the problems worth spending was awarded the first Chern Medal for asked Ernest’s wife, who was a friend of director, and he retired in 1999. He still lives time on. He has supervised more than forty lifetime achievement by the International his from Montreal, to ask her father-in-law in Manhattan. Ph.D. students and is an excellent lecturer Mathematical Union and the Chern Medal for advice about where to do graduate In the 1950s the Courant Institute and writer. Foundation. studies in theoretical physics. Richard was rapidly becoming one of the US’s top Courant responded that he should study research centres for applied mathematics, mathematics at his department at NYU. on a par with more established universities,

10 11 A glimpse of the Laureates’ work to how the surface is embedded in the three- Whereas the one-dimensional version of dimensional space. Curvature is an intrinsic Nash’s theorem is rather intuitive, the two- property of the surface, i.e. a property that dimensional version is more or less counter- Never change belongs to the surface by its very nature. intuitive, as the following illustration shows. Consequently it has to be preserved by any Start with a piece of paper and turn it into a a given distance ... isometric embedding. cylindrical shape. This is easy; the next step In the first embedding theorem of John is the hard part: to turn the cylinder into a Arne B. Sletsjøe F. Nash, Jr., published in 1954, he proves doughnut-shaped surface without stretching that any Riemannian manifold can be or tearing the paper. Intuitively this seems isometrically embedded in Euclidian space to be impossible. The outer circumference by a C1-map. The striking point of a curve of the doughnut is much longer than the version of this theorem is that any curve in inner, but in the original cylinder they are of the plane can be arbitrarily prolonged in a the same length. By Nash’s theorem this is Neurons are not evenly distributed in the prescribed Gauss curvature or Riemannian smooth way, without self-crossing and as never the less possible, at least theoretically. human body. Some parts of the body, like metric, solved the classical problems of close to the original curve as we want. The Nash proved the theorem in 1954, but it the hands, face and tongue are much more Minkowski and Weyl.” prolonged curve looks like the path of the was only in 2012 a multidisciplinary team sensitive to sensations than other parts. The A long time before spacecraft provided front wheel of a bicycle climbing a steep hill, in France, the HEVEA project, was able to body has the highest density of neurons us with images of the earth, our forefathers while the rear-wheel tracks out the original image the process where the cylinder is in those parts. A function that measures concluded that our planet is round. They curve. By increasing the frequency of twists bent into a doughnut, in an isometric way. the density of neurons is an example of based this knowledge on observations the cyclist can increase the difference The images in figure 2 illustrate the process; what mathematicians call a metric. An- done on the surface of the earth. By between the length of the front-wheel path the paper is warped by an infinite sequence other example of a metric is the so-called performing smart observations and correct and the rear-wheel path. Unlike the surface of waves, piling up to a doughnut surface in Euclidean metric, named after the ancient measurements, they were able to conclude case, curvature of a curve does not have to such a way that the original piece of paper Greek mathematician Euclid. The Euclid- that the earth could not be flat. If you fix be preserved by an isometric embedding. is kept intact. ean metric measures ordinary distances a point on a flat surface and you walk a between points and the area of any region circular path at a given distance R, the path An embedding theorem in mathematics concerns itself An isometric embedding is an embedding where all of a surface. In a paper from 1916 Hermann should be 2 R long. But if you measure with the extent to which it is possible to put one object distances between points are preserved. Distances are π into another, without “destroying” the objects. measured in the surface, so making a cylindrical shape Weyl asked the following question: Is it carefully on the earth’s surface you will out of a piece of paper is an isometric operation, whereas always possible to realise an abstract metric find that the perimeter is a little shorter. A flattening out a sphere is a non-isometric operation. on the 2-sphere of positive curvature by an theoretical computation then tells you that isometric embedding in R3? If you think of the earth’s surface has positive curvature, the neuron density metric as Weyl’s abstract i.e. locally it looks like a sphere. metric and the human body as the 2-sphere, The fact that it is possible to say then the weird body in figure 1 illustrates anything about the curvature, merely by the positive answer to Weyl’s question. The observations performed on the surface, was different sizes of the various body parts formulated by the great mathematician Carl correspond to the neuron density. Friedrich Gauss in 1827, in what is called The connection between Weyl’s Gauss’ Theorema Egregium, the remarkable question and the work of Luis Nirenberg is theorem. The theorem says that the Gaussian emphasized in the citation of the Abel Prize: curvature of a surface can be determined Figure 1: The different size of the parts of Figure 2: Images of an isometric embedding “Nirenberg, with his fundamental embedding entirely by measuring distances and angles the body reflects the density of neurons. of a flat torus in 3R . theorems for the sphere S2 in R3, having on the surface itself, without further reference Source: Natural History Museum, London Source: HEVEA Project/PNAS

12 13 About the Abel Prize

The Abel Prize is an international award at the Institute for Mathematics and its for outstanding scientific work in the field Applications in Minnesota, and The Bernt of mathematics, including mathematical Michael Holmboe Memorial Prize for aspects of computer science, mathematical excellence in teaching mathematics in physics, probability, numerical analysis, Norway. In addition, national mathematical scientific computing, statistics, and also contests, and various other projects and applications of mathematics in the sciences. activities are supported in order to stimulate The Norwegian Academy of Science interest in mathematics among children and Letters awards the Abel Prize based and youth. upon recommendations from the Abel Committee. The Prize is named after the — exceptional Norwegian mathematician Niels Henrik Abel (1802–1829). According to the Call for nominations 2016: statutes of the Abel Prize, the objective is The Norwegian Academy of Science and both to award the annual Abel Prize, and Letters hereby calls for nominations for to contribute towards raising the status of the Abel Prize 2016, and invite you (or mathematics in society and stimulating the your society or institution) to nominate interest of children and young people in candidate(s). Nominations are confidential mathematics. The prize carries a cash award and a nomination should not be made of 6 million NOK (about 700,000 Euro or known to the nominee. about 800,000 USD) and was first awarded in 2003. Among initiatives supported are Deadline for nominations for the Abel Prize the Abel Symposium, the International 2016 is September 15, 2015. Please consult Mathematical Union’s Commission for www.abelprize.no for more information Developing Countries, the Abel Conference

14 Yakov Sinai receives the Abel Prize from HRH Crown Prince Haakon in the University Aula, Oslo, May 2014.

The Abel Prize Laureates

2014 2013 2012 2011 2010 2009 Yakov G. Sinai Pierre Deligne Endre Szemerédi John Milnor John Torrence Tate Mikhail Leonidovich Gromov “for his fundamental contributions “for seminal contributions to “for his fundamental contributions “for pioneering discoveries in “for his vast and lasting impact to dynamical systems, ergodic algebraic geometry and for their to dis­ crete mathematics and topology, geometry and algebra.” on the theory of numbers.” “for his revolutionary contributions theory, and mathematical transformative impact on number theoretical computer science, to geometry.” physics.” theory, representation theory, and in recognition of the profound and related fields.” and lasting impact of these contributions on additive number theory and ergodic theory.”

2008 2007 2006 2005 2004 2003 John Griggs Thompson Srinivasa S. R. Varadhan Lennart Carleson Peter D. Lax Sir Michael Francis Atiyah Jean-Pierre Serre and Jacques Tits and Isadore M. Singer “for his fundamental contributions “for his profound and seminal “for his groundbreaking “for playing a key role in shaping “for their profound achievements to probability theory and in contributions to harmonic contributions to the theory and “for their discovery and proof the modern form of many parts in algebra and in particular for particular for creating a unified analysis and the theory of application of partial differential of the index theorem, bringing of mathemat­ics, including shaping modern group theory.” theory of large deviations.” smooth dynamical systems.” equations and to the computation together topology, geometry topology, algebraic geometry of their solutions.” and analysis, and their outstand ­ and number theory.” ing role in building new bridges between mathematics and theoretical physics.”

18 19 Programme Abel Week 2015

May 18 Abel Banquet at Akershus Castle in © Peter Badge/Typos 1 in coop. with the HLF – all rights reserved 2015. | © NYU Photo Bureau: Hollenshead Front page photos, left to right: honor of the Abel Laureates — Hosted by the Norwegian Government Holmboe Prize Award Ceremony (by invitation only from the Norwegian The Minister of Education and Research Government) presents the Bernt Michael Holmboe Memorial Prize for teachers of mathematics at Oslo Cathedral School — May 20 — Wreath-laying at the Abel Monument The Abel Lectures by the Abel Prize Laureates in the Palace Laureate Lecture, Science Lecture, and Park other lectures in the field of the Laureates’ work at Georg Sverdrups Hus, Aud. 1, May 19 University of Oslo — — The Abel Party Abel Prize Award Ceremony at The Norwegian Academy of Science and His Majesty The King presents the Abel Letters (by invitation only) Prize in the University Aula, University of Oslo — Reception and interview with the Abel May 21 — Laureates Laureate Lectures and events for school Science writer Vivienne Parry interviews the children in Bergen Abel Laureates at Det Norske Teatret Programme at Festplassen, and Laureate lectures at the University of Bergen

The Norwegian Academy Press contact: For other information: of Science and Letters Anne-Marie Astad Anette Burdal Finsrud [email protected] [email protected] +47 22 12 10 92 +47 415 67 406

Register online at: www.abelprize.no from mid-April, or contact [email protected]