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.