TWO outstanding mathematicians of the 20th century, the Americans John Forbes Nash Jr of Princeton University and Louis Nirenberg of the Courant Institute of Mathematical Sciences, New York University (NYU), have been chosen for the 2015 Abel Prize, the Nobel equivalent for mathematics. Announcing the winners on March 25, the Norwegian Academy of Sciences and Letters, which has been awarding this prestigious prize since 2003, said in its citation that the two were being awarded for their “striking and seminal contributions to the theory of non-linear *partial differential equations* and its applications to *geometric analysis*”.

The Abel Prize (“Science of chance”, *Frontline*, April 20, 2007) recognises contributions of extraordinary depth and influence in mathematical sciences and comes with a cash award of six million Norwegian kroner (about €700,000 or $750,000), which this year the two winners will share equally. The awards ceremony will be held on May 19 in Oslo and King Harald of Norway will give away the prize.

Of the two, Nash’s name is a familiar one even to the general public thanks to the 2001 film about him, *A Beautiful Mind*, loosely based on Sylvia Nasar’s bestseller biography of the same name, which was published in 1998, four years after Nash won the Nobel Prize in Economic Sciences. Ron Howard directed the film and Russell Crowe played Nash. The Nash of the film version, of course, differed from the real-life Nash significantly, which the film-makers acknowledged, saying that the film was not meant to be a true representation.

Hollywood’s interest in Nash’s life obviously stemmed from the attractive storyline (from the box-office perspective) that could be built around it: a mathematics wizard with an extremely complex character and odd social behaviour who is diagnosed with paranoid schizophrenia and goes on to win the Economics Nobel for his work in game theory as a graduate student in 1949, which today is used widely in applied economics. The film did well at the box office. Unsurprisingly, however, the film focussed only on his early results in game theory and totally ignored his exceptional research into geometry and partial differential equations (PDEs), which for the mathematical community are his most important contributions. It is for this work the 86-year-old Nash has been awarded the Abel Prize this year along with the 90-year-old Nirenberg.

Although the two did not formally collaborate on any of their papers, it is the fact that they influenced each other greatly with their work during the 1950s and a common thread runs through their works, which is what led them to be considered together for the prize. The two italicised phrases in the citation are central to their work: PDEs and geometric analysis. The year’s Abel Prize is also recognition of the interconnections that have emerged between the two fields subsequent to their contributions.

**What are PDEs?**

What are PDEs? These are equations involving rates of change of varying quantities that arise in the description of basic phenomena in physics, chemistry, biology and other sciences. The mathematical concept underlying PDEs is basically calculus. The history of calculus dates back to the 17th century when Isaac Newton and Gottfried Leibniz independently developed the technique to calculate the instantaneous rates of change of variables such as speed and acceleration that Newton needed to write the equations of motion. The instantaneous rate of change of a variable is called its “differential” and equations involving such quantities are called differential equations.

When a mathematical quantity depends on two variables, say, temperature depending on position and time as it does in the equation describing heat flow, one talks of “partial differentials”, which are the rates of change of the quantity with respect to one variable or the other. PDEs arise when systems depend on several variables simultaneously. The heat flow equation is, thus, a PDE as are the Maxwell’s equations of electromagnetism, the wave equation describing the motion of waves, the Navier-Stokes equations describing the dynamics of fluid flow, and the Schrödinger equation, which is about quantum mechanics. The most interesting PDEs are the ones that are non-linear, for example, the Navier-Stokes equations. The importance of PDEs can be seen from the fact that of the seven one-million-dollar Millennium Problems of the Clay Mathematics Institute, three are on, or related to, PDEs.

Geometry is the description of points, lines and planes, or surfaces in general, and the study of their behaviour. Euclidean geometry, as described in the great Greek mathematician Euclid’s classic *Elements*, deals with planar surfaces only where the sum of the angles of a triangle is always 180°. But one knows that the sum of the angles of a triangle drawn on a sphere is always greater than 180° and when drawn on the inside of a saddle-like surface is always less than 180°. This leads to the concept of curvature of surfaces. For a Euclidean flat plane, the curvature is zero. When the surface folds on itself, like in the case of a sphere, the curvature is positive, and it is called a convex surface. When the surface spreads out like a parabolic dish antenna or the inside of a saddle, the curvature is negative, and it is a concave surface. When the curvature is non-zero, Euclidean rules of angles are not valid.

On the basis of their real-life experience of living in a three-dimensional Euclidean space, historically, early mathematicians considered surfaces 2D subspaces embedded in this exterior 3D Euclidean space; that is, surfaces were defined and studied *extrinsically*. But, in 1827, the famous German mathematician Carl Friedrich Gauss introduced new notions and definitions and proved theorems about curvature that elevated the concept of a curvature to a rigorous mathematical *intrinsic* property of surfaces.

According Gauss’ famous *Theorema Egregium* (Latin for remarkable theorem), the (“Gaussian”) curvature of a surface was an intrinsic, rather than an extrinsic, property of the surface irrespective of its position (the manner of its embedding) in a 3D space. That is, a being confined to this 2D surface can determine the curvature of the surface just by measuring distances and angles on the surface without looking out into the 3D world in which the surface is embedded. This is in contrast to the case of a simple 1D curve, where only the length of the curve is intrinsic while its curvature is extrinsic. Think of the curve as an inelastic rope. While its length remains fixed, its curvature depends on how the rope is coiled. Gauss’ theorem is remarkable because the curvature of a surface was originally defined extrinsically.

The intrinsic curvature of a surface, for instance, can be measured by checking how closely the arc lengths of circles correspond with the value 2πr in planar Euclidean geometry: if it is smaller ta the surface, it is positively curved; if it is greater, it is negatively curved; and if it is the same, it is a flat surface. That is, human beings could have determined that the earth is spherical, and not flat, by simply making measurements on the earth’s surface alone without looking at the earth from outer space or looking out into the sky at the motion of stars. Further, according to Gauss’ theorem, this (Gaussian) curvature is invariant with respect to (smooth) deformations of the surface if they are such that the distances between two points on the surface are preserved (isometry).

Influenced by Gauss’ work on surfaces in Euclidean 3D space, his student Bernhard Riemann (1822-66) took his mentor’s work to a higher abstract level and introduced in 1854 what is today called Riemannian geometry. It was a generalisation of Euclidean geometry and was a new highly abstract way of describing surfaces in a purely intrinsic manner, disconnected from any exterior world or extrinsic property. That is, a Riemannian surface is a geometric object without any reference to its embedding in an ambient space. The surface is defined in abstraction purely in terms of rules for measuring distances and angles.

This abstraction also enabled Riemann to generalise this geometry from 2D to an arbitrary number of dimensions. These revolutionary ideas marked the birth of modern geometry, where ideas of Euclidean geometry, such as distances, angles, curvature, etc., get abstracted and generalised, and are treated in terms of *differentials*. Hence, the subject is today referred to as differential geometry.

Geometric analysis refers to the application of the theory of PDEs to geometry and, conversely, to the use of geometrical methods in the study of PDEs. It includes the study of problems involving curves and surfaces, or domains with curved boundaries, and also the study of Riemannian spaces in arbitrary dimensions. These developments had an immediate impact on the evolution of geometric and physical ideas in the 20th century. For example, Einstein’s theory of general relativity, which was formulated in 1915, is based on Riemann’s ideas and the geometry is of a 4D curved space-time.

As geometry in its basic approach is visual, intuitive and concerned with the larger picture, it has had a much longer history than PDEs, which have to do with details about small variations and had to wait for the evolution of appropriate mathematical tools. And yet the works of the Abel laureates show that geometry and PDEs are intimately linked and can feed into each other’s development and understanding. Mathematical research over the past decades has shown how PDEs can play an important role in the study of geometric objects in arbitrary dimensions as well.

One of the important contributions of the two laureates concerns the so-called “embedding theorems”. As mentioned earlier, Riemann surfaces do not exist in the real 3D world. These are abstractly defined geometric objects that are difficult to visualise in three Euclidean dimensions. The question that mathematicians posed was whether these Riemann surfaces could be embedded in Euclidean space. Embedding means every point in the abstract space is mapped to a point on a (2D) surface in the real world. In isometric embedding, the distance between any two points on the abstract surface is preserved on the mapped surface as well. Isometric embedding enables a concrete visualisation of what is actually an abstract geometrical concept.

“Nash’s embedding theorems,” the Abel Committee’s citation says, “stand among the most original results in geometric analysis of the 20th century… [and]… Nirenberg, with his fundamental embedding theorems for the [2D] sphere… [in 3D Euclidean space], having prescribed Gauss curvature or Riemannian metric, solved the classical problems of [Hermann] Minkowski and [Herman] Weyl….” The notions of a “prescribed Gauss curvature” or a “prescribed Riemannian metric” arise when dealing with the generalised concepts of curvature and distance in abstract spaces in the Riemannian sense.

Consider, for example, a sphere of radius R. The curvature of a sphere is positive everywhere and is equal to 1/R. Minkowski (1804-1909) asked in 1903 whether one could define a new surface whose curvature at every point was given by some function defined on the surface of the sphere that was positive at every point. Such a function could be, for example, the average rainfall on the surface of the earth. Now, one can construct a new surface in the sense of Minkowski, whose curvature is positive everywhere, by flattening the sphere in the parts where the “rainfall function” is less than the constant curvature 1/R and increasing the curvature of those areas where the function is greater by appropriate amounts.

**Weyl’s problem**

In 1916, Weyl (1885-1955) posed a related problem: Is it always possible to realise an abstract metric on the 2D sphere of positive curvature by an isometric embedding in 3D Euclidean space? The word “metric” in the above statement needs an explanation. It arises in the generalisation of the idea of distance and is something like a scale factor. In Euclidean space, the metric measures ordinary distances between any two points and this scale factor is the same everywhere and in all directions. In more generalised abstract spaces, the factor need not be the same everywhere and even at a given point it can be different in different directions.

This generalised concept can be illustrated by an example given in the background document issued by the Abel Foundation. Consider the surface as part of a landscape. One can introduce a metric on this space in terms of differences in walking speeds in different directions at every point, which is more general than the rainfall example. In marshy parts, the speed will be lower than on dry patches. The time taken to move between any two points on the surface can be regarded as the distance between those points. A plane with the Euclidean metric has zero Gaussian curvature, as explained before, while the same plane endowed with the “how-difficult-is-it-to-walk” metric described above would be rather curved.

The Weyl problem can now be stated as, given such a metric on a sphere, can the sphere be deformed such that on the deformed sphere the ordinary distance corresponds to the distance measured by the new abstract metric? Following the example the Norwegian mathematician Arne B. Sletsjoe provided in the background document, consider the neurons in the human body; they are not evenly distributed. The density is higher in body parts that are more sensitive to sensation, such as the hands, the face or the tongue. A function that measures the density of neurons in different directions at each point is an example of a metric. The neuron density can be thought of as Weyl’s abstract metric and the human body the 2D sphere. Then, the weird body in Figure 1 illustrates the positive answer to Weyl’s poser. The different sizes of the various parts of the body in the figure correspond to the neuron density.

Nirenberg’s solutions to these open problems were published in 1953 as a paper titled “The Weyl and Minkowski Problems in Differential Geometry in the Large”. The solution to the Weyl problem actually formed part of his PhD thesis, for which he worked under James Stoker. The thesis itself was a window to Nirenberg’s future interests: PDEs, elliptic PDEs in particular. (In elliptic PDEs, the coefficients satisfy an inequality. These have applications in nearly all areas of mathematics and in numerous areas of physics.)

“Weyl had solved it partly, and what I did was complete the proof,” Nirenberg said in an interview to the Notices of the American Mathematical Society in 2001. “Weyl had worked on it… and had made some crucial estimates. One needed some more estimates before one could finish the problem. What I did was to get the additional estimates, essentially using ideas of C.B. Morrey. Morrey’s work was a very big influence on me….” Nirenberg proved the problem by reducing it to a non-linear PDE, which is also an elliptic PDE.

In proving an embedding theorem, one needs to equate how one moves around the abstract space with how one moves around Euclidean space, and this naturally gives rise to PDEs. These solutions were important for two reasons: one, the problems were representative of a developing field and, two, the methods that Nirenberg evolved laid the basis for further applications. “Nirenberg’s solution of the Minkowski problem was a milestone in global geometry,” says the background document.

But Nirenberg did not follow up on his thesis work on the embedding problem in greater generality. “The work on the embedding problem involved non-linear partial differential equations. That’s how I got into partial differential equations. After that, I worked essentially in partial differential equations connected to other things,” Nirenberg said in the interview. As the mathematician J. Mawhin pointed out, ellipticity is a key word in Nirenberg’s work. More than one-third of his papers contain the word “elliptic” in their titles. “There is hardly any aspect of these equations that he has not considered,” said Mawhin. Nirenberg has contributed in a large measure to the now well-developed theory of elliptic PDEs.

Inequalities too figure extensively in Nirenberg’s work, and there are several important inequalities associated with his name. His attraction for inequalities is due to his long association with Kurt Friedrichs at the Courant Institute which Nirenberg joined after completing physics and mathematics at McGill University in 1945. Although he wanted to do physics, he ended up doing mathematics on the advice of Richard Courant. Nirenberg has said that Friedrichs was a major influence on him and that Friedrichs’ views on mathematics shaped his perspective to a great extent.

Nash’s interest in mathematics began at the age of 14, when he was inspired by E.T. Bell’s *Men of Mathematics*. He apparently succeeded in proving for himself Pierre de Fermat’s results stated in the book. He joined Bluefield College in 1941 and studied mathematics and sciences, particularly chemistry, which was his favourite, and demonstrated extraordinary abilities in mathematics. With a scholarship that he won in the George Westinghouse Competition, he joined the Carnegie Institute of Technology (now Carnegie-Mellon University) in 1945 to do chemical engineering. He also took courses in calculus and relativity where his mathematical talents were spotted by John Synge and others of the mathematics department who encouraged him to take up mathematics.

Nash obtained a B.A. and an M.A. in mathematics in 1948 and on Synge’s advice moved to Princeton. He obtained his doctorate from Princeton in 1950 for his work in game theory, but even before writing his thesis, Nash had obtained some important results in mathematics. In 1952, he was appointed an assistant professor at the Massachusetts Institute of Technology, and in early 1953, the mathematician Warren Ambrose at MIT wrote this of him to Paul Halmos at Chicago University: “He recently heard of the unsolved problem about embedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem was sufficiently worthwhile to justify his efforts…. He’s a bright guy but conceited as hell, childish as [Norbert] Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y.” Ambrose and Nash apparently had rubbed each other the wrong way for a while, and it was Ambrose who had told Nash: “If you’re so good, why don’t you solve the embedding theorem for manifolds.” And Nash went ahead and did it!

Sylvia Nasar’s biography of Nash says it was Nirenberg who suggested the problem to Nash. But in his interview Nirenberg said: “I don’t remember. I would say it’s likely because it was a problem that I was interested in and had tried to solve… but I’m not sure.” But of Nash, Nirenberg had this to say: “About 20 years ago, somebody asked me, ‘Were there any mathematicians you would consider as geniuses?’ I said, ‘I can think of only one, and that’s John Nash.’…When he was hanging around Courant… he would come around and ask questions like, ‘Do you think such and such inequality might be true?’ Sometimes inequalities weren’t true. I wasn’t sure he was getting anywhere. But then in the end, he did it. He had a remarkable mind. He thought about things differently from other people.”

Nash’s isometric embedding theorems are more general. He proved that any Riemannian surface can be made concrete (by isometric embedding) in a Euclidean world though sometimes one may need to invoke more than the familiar 3D world to do it. He proved two theorems, known as C(1) theorem and C(k) theorem, which were published in 1954 and 1956 respectively. The former is said to have a very simple proof but leads to some very counter-intuitive conclusions. While reviewing the C(1) paper, the mathematician Shiing-Shen Chern had remarked that it “contains some surprising results”. The proof of the second is apparently very technical but does not lead to any surprising results. The first one was also extended by Nicholas Kuiper in 1954 itself, and the theorem is known as the Nash-Kuiper theorem.

One of the counter-intuitive results that follow from the C(1) theorem can be illustrated by the following. Consider a square whose sides are pairwise identified. This is known as a square flat torus. An imaginary 2D being living in this space would exit the upper side and reappear from the lower side, and similarly, each time it disappeared from the left, it would reappear from the right (Figure 2). This is like some computer games where characters exit from one side and re-enter from the opposite side.

To visualise this imaginary world concretely, one will try and “embed” this in the real 3D world. Take a square piece of some soft material. Now one can lift the upper and lower edges in the third dimension, bend them and join them to get a cylinder. This is easily done. One similarly bends the right and left edges too in the third dimension and joins them. With a soft deformable material, this can be done though with some difficulty. In any case, one can certainly imagine the resulting object, which would look like a bicycle wheel’s inner rubber tube. That is, the abstract space of a 2D flat torus is nothing but a ring buoy or a hollow doughnut in the 3D world (Figure 3a). But this embedding is not isometric as it does not preserve distances (Figure 3b). For example, the inner circumference of the buoy (originally the vertical in the square) is smaller than the outer circumference (originally the horizontal in the square).

Isometric embedding, thus, seems impossible and so it was believed until Nash (and Kuiper) proved that such a representation, which does respect distances, indeed exists. But, even in mathematics, proving and actually showing by construction are two different things. This had to wait until 2012 when a team of French mathematicians undertook the HEVEA project and was actually able to demonstrate that a flat torus can indeed be represented as a doughnut in the 3D world in an isometric way.

While Nash and Kuiper had proved the existence of such a representation, their proofs do not provide any explicit constructional procedure. This became possible with the work of Mikhail Gromov, a Russian-French mathematician and a 2009 Abel laureate, who explored the works of Nash and Kuiper to come up with a new method called “convex integration”, which the French group adapted to devise an algorithm to implement the embedding on a computer and visualise it through images (Figure 4).

The works of Nash and Kuiper had actually puzzled the mathematical community. They showed the existence of objects with complex regularity or smoothness properties: they were both smooth and rough at the same time. The computer realisation of the Nash theorem has revealed that such a surface that is both smooth and jagged, but infinitely broken, is indeed possible. If one zooms in on the images, what one sees is an infinite sequence of waves at smaller and smaller scales. Each ripple appears smooth when viewed alone but the piling up of such ripples creates an object that is rough. Thus, the Nash theorem’s realisation has revealed a new class of surfaces with unsuspected geometry.

Like Nirenberg, the technique Nash invented to solve the elliptic PDE in his embedding theorem was so innovative and useful that his results have had an impact within the field of PDEs though the problem was originally a geometric one. The HEVEA realisation of his theorem has also opened up new avenues of research in applications of mathematics to other fields by providing new tools of determining atypical solutions to PDEs, say in physics and biology, through the technique of convex integration.

The Abel award is actually for the body of the works of Nirenberg and Nash that is much more than their embedding theorems. But these results have provided a foundation for new avenues to open and other mathematicians to explore the full richness of PDEs. For instance, the Russian mathematician Grigori Perelman combined the two fields of PDEs and geometry in his proof in 2006 of the Poincare Conjecture, one of the most famous unsolved problems in mathematics.

Please Email the Editor