2010 Chern Medal Awarded On August 19, 2010, the first and nonlinear partial differential equations (PDEs) Chern Medal Award was pre- and related areas of analysis, the basic mathemati- sented at the opening ceremo- cal tools of modern science. PDEs arise in physics nies of the International Con- and geometry when systems depend on several gress of Mathematicians (ICM) in variables simultaneously, and the most interest- Hyderabad, India. The awardee ing ones are nonlinear. The importance of PDEs is is Louis Nirenberg of the Cou- clear in the fact that, of the seven million-dollar rant Institute of Mathematical Millennium Problems posed by the Clay Mathemat- Sciences, New York University. ics Institute, three are in or are related to PDEs. He was honored “for his role in Nirenberg’s work in PDEs is deep and fundamen- the formulation of the modern Photograph courtesy of ICM/IMU. tal. He developed intricate connections between theory of nonlinear elliptic par- Louis Nirenberg analysis and differential geometry and applied tial differential equations and them to the theory of fluid phenomena and other for mentoring numerous students and postdocs physical processes. in this area.” Nirenberg’s thesis concerned a fundamental The Chern Medal Award was instituted for 2010 issue in geometry: the solution to an embedding in memory of the outstanding mathematician problem in differential geometry that Hermann Shiing-Shen Chern. It will be awarded jointly with Weyl had posed around 1916. Given a Riemannian the Chern Medal Foundation (CMF) to an individual metric on a unit sphere with positive Gauss curva- whose lifelong outstanding achievements in the ture, can you embed this two-sphere isometrically field of mathematics warrant the highest level of in three-space as a convex surface? The thesis itself recognition. was a window to Nirenberg’s main interests—PDEs, Shiing-Shen Chern devoted his life to mathemat- and especially elliptic PDEs. In giving a positive ics, both in active research and education, and answer to Weyl’s question, Nirenberg used ideas to nurturing the field whenever the opportunity due to Charles Morrey. He solved the problem by arose. He obtained fundamental results in all the major aspects of modern geometry and founded reducing it to a problem in nonlinear PDE. This the area of global differential geometry. Chern equation is also an elliptic PDEs. (In elliptic PDEs, exhibited keen aesthetic tastes in his selection of the coefficients satisfy a positivity condition, and problems, and the breadth of his work deepened these have applications in almost all areas of the connections of modern geometry with different mathematics, as well as numerous applications in areas of mathematics. physics. As with a general PDE, an elliptic PDE may The award consists of a medal and a monetary have nonconstant coefficients and be nonlinear. prize of US$500,000. It is required that half of The basic example of an elliptic PDE is Laplace’s the award be donated to organizations of the equation.) recipient’s choice to support research, education, As mathematician J. Mawhin has pointed out in outreach, or other activities to promote mathemat- his recent tribute to Nirenberg, ellipticity is a key ics. Chern was generous during his lifetime in his word in Nirenberg’s mathematical work. More than personal support of the field, and it is hoped that one-third of Nirenberg’s articles contain the word this philanthropy requirement for the promotion “elliptic” in their titles, and a much larger fraction of mathematics will set the stage and the standard of them deal with elliptic equations or systems. for mathematicians to continue this generosity on “There is hardly any aspect of those equations he a personal level. has not considered,” Mawhin said. Despite their Louis Nirenberg is undoubtedly one of the variety, elliptic PDEs have a well-developed theory outstanding analysts of the twentieth century. to which Nirenberg contributed in large measure. His work has had major influence in the develop- In 1953, the first year in which he published, he ment of different areas of mathematics and their published three further papers, two of which applications. In particular, he has been a leader in involved nonlinear elliptic PDEs, to which he re- most of the developments in the theory of linear turned in later years. 1472 NOTICES OF THE AMS VOLUME 57, NUMBER 11 Through the next twenty years, with several As mentioned earlier, Nirenberg, a pioneer in collaborators, he developed the theory of elliptic nonlinear PDEs, has turned his attention again at equations satisfying certain regularity criteria that various stages to fully nonlinear equations and he had formulated. Although proofs for the exis- made striking breakthroughs. An example is the tence and uniqueness of weak solutions to elliptic series of papers on the existence of smooth solu- problems were well known, Nirenberg addressed tions of equations of Monge-Ampère type, which the much more difficult question of how regular are nonlinear second-order PDEs with special (or well behaved) the weak solution is. (A weak symmetries, with Luis Caffarelli and Spruck. Niren- solution to a PDE is one for which the derivatives berg’s study of symmetric solutions of nonlinear appearing in the equation may not all exist but elliptic equations using moving plane methods, which nevertheless is deemed to satisfy the equa- with Basilis Gidas and Wei Ming Ni, and later with tion in a certain dual sense.) Today, Nirenberg’s Henri Berestycki, is an ingenious application of the method of differences for proving the interior and maximum principle. Mawhin has called Nirenberg boundary regularity is part of a graduate student’s a “Paganini of the maximum principle”. Thanks to education in PDEs. Nirenberg’s ideas and methods, this has developed A high point in his research in this area is the into a beautiful theory and has led to applications extension of this work, which was done in collabo- in combustion theory. ration with Shmuel Agmon and Avron Douglis on a The following famous quotations by Nirenberg priori estimates for general linear elliptic systems. are indicative of his approach to handling nonlin- This is one of the most widely quoted results in earity; in Nirenberg's view, the problem determines analysis. As Nirenberg has said, the objective was the method. In his invited lecture at the Stockholm to obtain “general estimates for general systems International Congress of Mathematicians (ICM under general boundary conditions”. The citation 1962), he said: “Most results for nonlinear prob- for the Steele Prize for Lifetime Achievement, lems are still obtained via linear ones, despite the fact that the problems are nonlinear and not which he received from the American Mathematical because of it.” In the same lecture he also said, Society (1994), said: “Nirenberg is a master of the commenting on someone’s result, “The nonlinear art and science of obtaining and applying a priori character of the equation is used in an essential estimates in all fields of analysis.” way; indeed, he obtains results because of the Essential questions about regularity for Navier- nonlinearity and not despite it.” Stokes equations, whose solutions determine fluid Nirenberg’s wide range of interests also includes motion, are still open, and it is one of the Clay differential geometry and topology, in which he Institute’s Millennium Problems. Among the best has made significant contributions. In harmonic results today is the Caffarelli-Kohn-Nirenberg es- analysis a function of bounded mean oscillation timate of the measure of the set of singularities. (BMO) is a real-valued function whose mean os- Nirenberg feels that the problem will be solved cillation is bounded (finite). Motivated by Fritz soon but that it will require more input from har- John’s earlier work in elasticity, Nirenberg, in as- monic analysis. sociation with John, investigated for the first time Building on earlier estimates of Alberto Calde- the topology of the space of such functions and rón and Antoni Zygmund, he and Joseph Kohn gave a precise definition for it. The space is also introduced the concept of a pseudodifferential sometimes called the John-Nirenberg space. The operator, a generalization of the idea of parametrix results were crucial for later work of Charles Fef- for a partial differential operator, which is useful ferman on Hardy spaces. This function space has in addressing the question of regularity of solu- subsequently been used in many parts of analysis tions to elliptic boundary value problems. They and in martingale theory. More recently, motivated were addressing a problem that involved singular by some nonlinear problems in physics, Nirenberg integral operators (integral operators that are returned to this field and, in collaboration with not mathematically defined at a point), and they Haim Brezis, investigated the space of functions needed facts about certain properties of singular with vanishing mean oscillation (VMO). The VMO integral operators, but these were not available in functions form a predual for the first Hardy space. the literature. So they developed what they needed, This function space is actually a subspace of BMO. and what resulted was the extremely useful notion Nirenberg and Brezis extended the “degree theory” of a pseudodifferential operator. This has helped in topology to mappings belonging to VMO, a result generate copious developments. Another impor- that took topologists by surprise. A fundamental tant work was the one with François Treves on question in the study of complex manifolds is: the solvability of general linear PDEs. Some other When is an almost complex structure given by a highlights are his research on the regularity of complex structure? A manifold is a topological free boundary problems with David Kinderlehrer space that can be locally described in terms of and Joel Spruck.