The leading lights at Courant were very much at the forefront of rapid progress, stimulated by World War II, in certain kinds of differential equations that serve as mathematical models for an immense variety of physical phenomena involving some sort of change. By the mid-fifties, as Fortune noted, mathematicians knew relatively simple routines for solving ordinary differential equations using computers. But there were no straightforward methods for solving most nonlinear partial differential equations that crop up when large or abrupt changes occur — such as equations that describe the aerodynamic shock waves produced when a jet accelerates past the speed of sound. In his 1958 obituary of von Neumann, who did important work in this field in the thirties, Stanislaw Ulam called such systems of equations “baffling analytically” saying that they “defy even qualitative insights by present methods.” As Nash was to write that same year, “The open problems in the area of non-linear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps, more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. It seems clear however that fresh methods must be employed.”
Nash, partly because of his contact with Norbert Wiener and perhaps his earlier interaction with Weinstein at Carnegie, was already interested in the problem of turbulence. Turbulence refers to the flow of gas or liquid over any uneven surface, like water rushing into a bay, heat or electrical charges travelling through metal, oil escaping from an underground pool, or clouds skimming over an air mass. It should be possible to model such motion mathematically. But, it turns out to be extremely difficult. As Nash wrote:
Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid. These are a non-linear parabolic system of equations. An interest in these questions led us to undertake this work. It became clear that nothing could be done about the continuum description of general fluid flow without the ability to handle non-linear parabolic equations and this in turn required an a priori estimate of continuity.
It was Louis Nirenberg, a short, myopic, and sweet-natured young protege of Courant’s, who had handed Nash a major unsolved problem in the then fairly new field of nonlinear theoty. Nirenberg, also in his twenties then, and already a formidable analyst, found Nash a bit strange. “He’d often seemed to have an internal smile, as if he was thinking of a private joke, as he was laughing at a private joke that he had never told anyone about.” But he was extremely impressed with the technique Nash had invented for solving his embedding theorem and sensed that Nash might be the man to crack an extremely difficult outstanding problem that had been open since the late 1950s:
He (Nirenberg) had recalled:
I worked in partial differential equations. I also worked in geometry. The problem had to do with certain kinds of inequalities associated with elliptic partial differential equations. The problem had been around in the field for some time and a number of people had worked on it. Someone had obtained such estimates much earlier in the 1930s in two dimensions. But the problem was open for almost thirty years in higher dimensions.
Nash had begun working on the problem almost as soon as Nirenberg suggested it, although he knocked on doors until he had been satisfied that the problem was as important as Nirenberg had claimed. Peter Lax, who was one of these he had consulted, had commented some time back: In physics, everybody knows the most important problems. They are well-defined. Not so in mathematics. People are more introspective. For Nash, though, it had to be important in the opinion of others.
Nash had started visiting Nirenberg’s office to discuss his progress. But, it was weeks before Nirenberg got any real sense that Nash was getting anywhere. “We would meet often. Nash would say, “I seem to need such and such an inequality. I think it’s true that…” Very often, Nash’s speculations were far off the mark. He was sort of groping. He gave that impression. I wasn’t very confident he was going to get through.
Nitenberg had then sent Nash around to talk to Lars Hormander, a tall, steely Swede who was then already one of the top scholars in the field. Precise, careful, and immensely knowledgeable, Hormander knew Nash, by reputation but had reacted even more skeptically than Nirenberg. “Nash had learned from Nirenberg the importance of extending the Holder estimates known for second-order elliptic equations with two variables and irregular coefficients to higher dimensions,” Hormander had recalled in 1997. “He came to see me several times. ‘What did I think of such and such an inequality?’ At first, his conjectures were obviously false. They were easy to disprove by known facts on constant coefficient operators. He was rather inexperienced in these matters. Nash did things from scratch without using standard techniques. He was always trying to extract problems…(from conversations with others). He had not the patience to study them.”
Nash had continued to grope, but with more success. “After a couple more times,” said Hormander, “he would come up with things that were not so obviously wrong.”
By the spring, Nash was able to obtain basic existence, uniqueness, and continuity theorems once again using novel methods of his own invention. He had a theory that difficult problems couldn’t be attacked frontally. He had approached the problem in an ingeniously roundabout manner, first transforming the nonllnear equations into linear equations and then attacking these by nonlinear means. “It was a stroke of genius,” said Peter Lax, who had followed the progress of Nash’s research closely. “I have never seen that done. I always kept it in my mind, thinking may be, it will work in another circumstance.”
(Note: Peter Lax is an earlier Abel Laureate).
Nash’s new result had gotten far more immediate attention than his embedding theorem. It had convinced Nirenberg, too, that Nash was a genius. Hormander’s mentor of the University of Lund, Lars Garding, a world class specialist in partial differential equations, had immediately declared, “You have to be a genius to do that.”
9. Relations of magnitude between real numbers.
It is plain, that, now that we have extended our conception of number, we are bound to make corresponding extensions of our conceptions of equality, inequality, addition, multiplication, and so on. We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of Chapter 1, part 1 blog. To do all this systematically would occupy considerable space/time, and we shall be content to indicate summarily how a more systematic discussion would proceed.
We denote a real number by a Greek letter such as , , ; the rational numbers of its lower and upper classes by the corresponding English letters a, A; b, B; c, C; …We denote the classes themselves by (a), (A),…
If and are two real numbers, there are three possibilities:
i) every is a b and every A a B; in this case, (a) is identical with (b) and (A) with (B);
ii) every a in a b, but not all A’s are B’s; in this case (a) is a proper part of , and (B) a proper part of (A);
iii) every A is a B, but not all a’s are b’s.
(These three cases may be indicated graphically on a number line).
In case (i) we write , in case (ii) , and in case (iii) . It is clear that, when and are both rational, these definitions agree with the ideas of equality and inequality between rational numbers which we began by taking for granted; and that any positive number is greater than any negative number.
It will be convenient to define at this stage the negative of a positive number . If
, (A) are the classes, which consitute , we can define another section of the rational numbers by putting all numbers in the lower class and all numbers in the upper. The real number thus defined, which is clearly negative, we denote by . Similarly, we can define
when is negative or zero; if is negative, is positive, It is plain also that . Of the two numbers and one is always positive (unless ). The one which is positive we denote by and call the modulus of .