A fundamental question in the study of complex manifolds is: When is an almost complex structure given by a complex structure? A manifold is a topological space that can be locally described in terms of such simpler, well-understood spaces as Euclidean. For a manifold to be a complex manifold (where operations with imaginary numbers can be defined), the existence of an almost complex structure is necessary, but not sufficient.
An almost complex manifold is a smooth manifold equipped with a structure that, roughly speaking, defines operations with imaginary numbers on each tangent space, the differentiable manifold attached to every point. That is, every complex manifold is an almost complex manifold, but not vice-versa. In 1957, Nirenberg with his student August Newlander proved a fundamental result that answered this question that had been open for many years.
According to Nirenberg, André Weil and Shiing-Shen Chern had drawn his attention to the problem of proving integrability condition for almost complex structures. Its resolution paved the way for its use in the study of many aspects of complex manifolds, particularly deformation theory. Although the problem is linear, Newlander and Nirenberg’s proof reduces the problem to a system of non-linear PDEs (partial differential equations), such that each equation involves derivatives with respect to only one complex variable. Nirenberg’s tremendous insight into the properties of PDEs and his unique ability to connect PDEs, analysis and geometry runs through all of Nirenberg’s work.
Inequalities have had a special attraction for Nirenberg and there are several important inequalities associated with his name and a very important result is the set of Garliardo-Nirenberg inequalities. The AMS citation called it a “minor gem”. Nirenberg’s love for inequalities comes from his long association with Kurt Friedrichs at the Courant Institute for Mathematical Sciences at New York University where Nirenberg ended after doing his under graduation with a major in physics and mathematics from McGill University in 1945.
Nirenberg said Friedrichs was a major influence on him and his views of mathematics very much formed his view. Though he had wanted to do his Ph.D under him, he ended up doing under Jim Stoker, which he finished in 1949. But Friedrichs’ influence is clearly visible in Nirenberg’s choice of problems, which are often drawn from physics. He does not distinguish between “pure” and “applied” mathematics, an attitude that has resulted from his entire career being spent at CIMS “where there is just mathematics and people are interested in pure and applied problems”, a heritage of Courant and Friedrichs.
Nirenberg is recognised for his excellent lectures and lucid expository writing. He has published over 185 papers and has had 46 students and 245 descendants according to the Mathematics Genealogy Project. In each of the last 10 years, top 15 cited papers in mathematics include at least two of Nirenberg, according to MathSciNet. The fact that nearly 90 per cent of Nirenberg’s papers are written in collaboration shows how Nirenberg, for over six decades, has shared his knowledge and mathematical insight with mathematicians from all over the world. He is also known to be very generous and enthusiastically presents the results of other mathematicians in his lectures and survey papers.
In his AMS interview, 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.”
(An excerpt from the work profile of Louis Nirenberg of Courant Institute of Mathematical Sciences, New York University. Nirenberg was awarded the Chern Medal, 2010, “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” at the International Congress of Mathematics, currently underway in Hyderabad, India)