John Nash’s unique approach produced quantum leaps in economics and maths


The American mathematician John Nash, who was killed on Saturday night in a car crash, was in Oslo five days ago to receive the Abel prize from the king of Norway. The GBP 500,000 Abel — which he shared with Louis Nirenberg — is considered a kind of maths version of the Nobel prize, which has no category for mathematics.

And yet, Nash is also a winner of the Nobel prize, the only person to share both accolades. “I must be an honorary Scandinavian,” he joked in March during the press conference that announced this year’s Abel laureates.

Nash is most famous for his research into game theory, the maths of decision-making and strategy, since it was this work that led to his being awarded the 1994 Nobel in economics. His fame also came from the 2001 Oscar-winning film A Beautiful Mind, in which he was played by Russell Crowe. The film, which turned him into probably the best known mathematician in the world, was based on the superb biography by Sylvia Nasar and charts his early career and then the struggle with schizophrenia that dominated most of his adult life.

Within the mathematics community, however, the work for which Nash was most admired — and for which he won the Abel — was not the game theory research but his advances in pure mathematics, notably geometry and partial differential equations .

The mathematician Mikhail Gromov once said: “What [Nash] has done in geometry is, from my point of view, incomparably greater than what he has done in economics, by many orders of magnitude. It was an incredible change in attitude of how you think.” Nash’s achievements in mathematics were striking not only because he proved deep and important results, but also because his career lasted only a decade before he was lost to mental illness.

Nash was born in 1928 in a small, remote town in West Virginia. His father was an electrical engineer and his mother a schoolteacher. He was an undergraduate at Carnegie Institute of Technology (now Carnegie Mellon University) in Pittsburgh and then did his graduate studies at Princeton, New Jersey. His PhD thesis, Non-Cooperative Games, is one of the foundational texts of game theory. It introduced the concept of an equilibrium for non-cooperative games, the “Nash equilibrium”, which eventually led to his economics Nobel prize.

Yet his mathematical interests soon lay elsewhere. He described his first breakthrough in pure mathematics, in his early 20s, as “a nice discovery relating to manifolds and real algebraic varieties”. His peers already recognised the result as an important and remarkable work.

In 1951, Nash left Princeton for MIT. Here, he became interested in the problems of “isometric embedding”, which asks whether it is possible to embed abstractly defined geometries into real-world geometries in such a way that distances are maintained. Nash’s two embedding theorems are considered classics, providing some of the deepest mathematical insights of the last century.

This work on embeddings led him to partial differential equations, which are equations involving flux and rates of change. He devised a way to solve a type of partial differential equation that hitherto had been considered impossible. His technique, later modified by J Moser, is now known as the Nash-Moser theorem.

In the early 1950s, Nash worked during the summers for the RAND Corporation, a civilian thinktank funded by the military in Santa Monica, California. Here, his work on game theory found applications in United States military and diplomatic strategy.

Perhaps Nash’s greatest mathematical work came from studying a mathematical puzzle that had been suggested to him by Louis Nirenberg. It concerned a major open problem concerning elliptic partial differential equations. Within a few months, Nash had solved the problem. It is thought that his work would have won him the Fields Medal — the most prestigious prize in maths, open only to those under 40 — had it not been solved at the same time by Italian mathematician Ennio De Giorgi. The men used different methods, and were not aware of each other’s work — the result is known as the Nash-De Giorgi theorem.

One of the many amazing aspects of Nash’s career was that he was not a specialist. Unlike almost all top mathematicians now, he worked on his own, and relished attacking famous open problems, often coming up with completely new ways of thinking. Louis Nirenberg once said, “About 20 years ago somebody asked me, “Are there any mathematicians you would consider as geniuses?” I said, “I can think of one, and that’s John Nash.” He had a remarkable mind. He thought about things differently from other people.”

In 1959, Nash began to suffer from delusions and extreme paranoia. For the next 40 years or so he was only able to do serious mathematical research in brief periods of lucidity.

Remarkably, however, he gradually improved and his mental state had recovered by the time he won the Nobel in 1994.

Nash showed such resolve and stamina in his mathematical work and in recovering from his mental illness, that his death in a taxi crash on the New Jersey turnpike seems all the more pointless and tragic.

May his soul rest in peace.

NeuroBreak: Thought Controls Prosthetic, Loss of a Beautiful Mind


A paralyzed patient was able to control a robotic arm with the help of a neural implant that tapped into regions of the brain associated with intention rather than motor skills.

John Nash, the Princeton mathematician whose struggle with schizophrenia was depicted in the movie “A Beautiful Mind,” and his wife, Alicia, died in a car crash on the New Jersey Turnpike on Memorial Day weekend.

U.S. appeals court won’t let Actavis take the Alzheimer’s drug Namenda off the market when generics hit, foiling its plans to switch all patients over to an extended-release version of the drug.

NFL players who lose consciousness when they get a concussion may be at greater risk of cognitive impairment, according to a JAMA Neurology study.

Two studies in JAMA confirm that beta amyloid levels can help predict development of Alzheimer’s disease, and that plaque buildup can occur 20 to 30 years before symptoms of dementia appear.

Depression may be an early sign of Parkinson’s, according to a study in Neurology.

Every world in a grain of sand: John Nash’s astonishing geometry – ScienceAlert


The mathematics of a beautiful mind.

As has been widely reported, John Forbes Nash Jr died tragically in a car accident on May 23 of this year. Many tributes have been paid to this great mathematician, who was made famous by Sylvia Nasar’s biography A Beautiful Mind and the subsequent movie based on that book.
Much has been said about Nash’s work on game theory. But less has been said about Nash’s other mathematical achievements. Many mathematicians who understand Nash’s work would agree, I think, that although his work in game theory had the most impact on other fields, Nash made other breakthroughs which were even more impressive.

Apart from game theory, Nash worked in fields as diverse as algebraic geometry, topology, partial differential equations and cryptography.

But perhaps Nash’s most spectacular results were in geometry. To honour Nash’s life, I would like to try to give a flavour of some of this work.

John Nash and pure mathematics

A great deal of Nash’s work was in the field of geometry. But this kind of geometry – differential geometry – is very different from the geometry learned at high school. It is not about trigonometry or Pythagoras, as found in secondary maths textbooks. Rather, it is about topics like surfaces, curvature and smoothness.

Like all pure mathematicians, Nash proved theorems: logical statements that are rigorous, precise and absolutely true, with no tolerance for vagueness. The world of pure mathematics is austere and often abstruse, but its claims to truth are eternal and absolute.

Well, that’s the theory at least. Breakthroughs in pure mathematics are often at the very limits of human understanding. It takes time, even for those in the field, to fully comprehend new developments.

Nash’s work was an extreme case. His papers could be chaotically presented, hard to follow and his approaches to problems were often unlike anything that had come before him, bamboozling students and experts alike. But he was almost otherworldly in his creativity.

While mathematical arguments are tightly constrained by the rigorous requirements of logic, Nash’s constructions and methods were wild. And nowhere was this more so than in his work on geometry.

Nash’s geometry

Take a flat sheet of paper. You can bend it, but without ripping it or creasing it, what shapes can you make? You can’t make a sphere, or even a section of a sphere, because a sphere is curved, while the paper is flat.

But you can make a cylinder. And even a cone, as you’ll know if you’ve ever seen a dunce’s hat. (This fact is also useful for making waffle cones, as shown below.)

image-20150527-11280-1fztjxw web

As it turns out, even though a cylinder or a cone looks curved, it is intrinsically flat. In an undergraduate course on differential geometry (such as the one I teach at Monash), one studies this intrinsic curvature, and it turns out that there are lots of flat surfaces.

image-20150527-25090-9hpk1f

These ideas were around for hundreds of years before Nash, but Nash took them much further.

The embedding problem

Nash took up the idea of ’embedding’ a surface: placing it into space without tearing, creasing or crossing itself. An embedding which does not distort the surface’s intrinsic geometry is ‘isometric’. In other words, the surfaces above are ‘isometric embeddings’ of the plane into 3-dimensional space.

The isometric embedding question can be asked not just for the plane, but for any possible surface: spheres, donuts (which mathematicians call tori to try to sound respectable) and many others.

As it turns out, there are surfaces that are so strongly curved or tangled up that they cannot be embedded into 3-dimensional space at all. In fact, they can’t even be embedded into 4-dimensional space.

But Nash showed that any surface can be embedded into 17-dimensional space. Extra dimensions, far from making the problem even more difficult, actually make it easier – giving you more room to embed your surface! Later on, Nash’s work was improved by others, and we now know that any surface can be embedded into 5-dimensional space.

However, surfaces are only 2-dimensional. And Nash was interested in surfaces of any possible dimension. These higher dimensional analogues of surfaces are known as ‘manifolds’.

Nash proved that you can always embed a manifold into space of some dimension, without distorting its geometry. With this momentous result, he solved the isometric embedding problem.

Nash’s proof of the isometric embedding problem came as a complete surprise to much of the mathematical community. His methods were revolutionary. The great mathematician Mikhail Gromov said that Nash’s work on the embedding problem struck him to be “as convincing as lifting oneself by the hair”. But after great effort, Gromov finally understood Nash’s proof: at the end of Nash’s lengthy argument, Gromov said, Nash “miraculously, did lift you in the air by the hair”!

Isometric embedding in action

Gromov went on to develop his own ideas, inspired by Nash’s work. He wrote a book – similarly renowned among mathematicians for its incomprehensibility, just like Nash’s work – in which he developed a method called ‘convex integration’.

Gromov’s method had several advantages. One is that it is easier to draw pictures of an embedding made with his convex integration method. Prior to Gromov, we knew isometric embeddings existed, and had wonderful properties, but had a very tough time trying to visualise them, not least because they were often in higher dimensions.

In 2012, a team of French mathematicians produced computer graphics of isometric embeddings using Gromov’s convex integration methods. They are extremely intricate, almost fractal-like, yet smooth. Some are shown below.

The world in a grain of sand

Nash’s work on the isometric embedding problem has many facets and has led to huge amounts of subsequent research.

One particularly amazing aspect is how isometric embeddings are constructed. Nash’s work, combined with subsequent work by Nicolaas Kuiper, showed that if you wanted to isometrically embed a surface in 3-dimensional space, it’s enough to be able to shrink it.

If you have a ‘shrunken’ embedding of your surface – that is, with all lengths decreased – then Nash and Kuiper show how you can obtain an isometric embedding of your surface just by adjusting your shrunken version a bit.

This sounds ridiculous. For instance, take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius. Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius. This type of ‘ruffling’ of the surface was reproduced in the French team’s computer graphics.

The French team considered taking a flat square piece of paper. Glue the top side to the bottom side, to get a cylinder. Now glue the left side to the right side. If you think about it, you might be able to see that you get a donut. But you’ll find the paper is now creased or distorted.

Can you embed it into 3-dimensional space without distortion? Nash and Kuiper say “yes”. Gromov says “use convex integration”. And the French mathematicians say “this is what it looks like”!

 

More pictures are available at the Project’s website.

But the mathematical theorem doesn’t just apply to tennis balls or donuts: the theorem holds for any manifold of any dimension. Any world can be contained in a grain of sand.

How did he do it?

Nash had a rare combination of genius and hard work. In her biography of Nash, Sylvia Nasar details his formidable intensity and effort spent working on the problem.

As is well known from the movie, Nash came to believe in outlandish conspiracy theories involving aliens and supernatural beings, as a result of his schizophrenia. When later asked why he, an extremely intelligent scientist, could believe in such things, he said those ideas “came to me the same way that my mathematical ideas did. So I took them seriously”.

And frankly, if my head told me ideas as accurate and as insightful as those needed to prove the isometric embedding theorem, I’d likely trust it on aliens and the supernatural too.

John Nash’s unique approach produced quantum leaps in economics and maths


John Nash.

Having solved some of the great theoretical problems and battled mental illness, the remarkable mathematician’s death in a car accident seems all the more tragic.

The American mathematician John Nash, who was killed on Saturday night in a car crash, was in Oslo five days ago to receive the Abel prize from the king of Norway. The GBP 500,000 Abel — which he shared with Louis Nirenberg — is considered a kind of maths version of the Nobel prize, which has no category for mathematics.

And yet, Nash is also a winner of the Nobel prize, the only person to share both accolades. “I must be an honorary Scandinavian,” he joked in March during the press conference that announced this year’s Abel laureates.

Nash is most famous for his research into game theory, the maths of decision-making and strategy, since it was this work that led to his being awarded the 1994 Nobel in economics. His fame also came from the 2001 Oscar-winning film A Beautiful Mind, in which he was played by Russell Crowe. The film, which turned him into probably the best known mathematician in the world, was based on the superb biography by Sylvia Nasar and charts his early career and then the struggle with schizophrenia that dominated most of his adult life.

Within the mathematics community, however, the work for which Nash was most admired — and for which he won the Abel — was not the game theory research but his advances in pure mathematics, notably geometry and partial differential equations .

The mathematician Mikhail Gromov once said: “What [Nash] has done in geometry is, from my point of view, incomparably greater than what he has done in economics, by many orders of magnitude. It was an incredible change in attitude of how you think.” Nash’s achievements in mathematics were striking not only because he proved deep and important results, but also because his career lasted only a decade before he was lost to mental illness.

Nash was born in 1928 in a small, remote town in West Virginia. His father was an electrical engineer and his mother a schoolteacher. He was an undergraduate at Carnegie Institute of Technology (now Carnegie Mellon University) in Pittsburgh and then did his graduate studies at Princeton, New Jersey. His PhD thesis, Non-Cooperative Games, is one of the foundational texts of game theory. It introduced the concept of an equilibrium for non-cooperative games, the “Nash equilibrium”, which eventually led to his economics Nobel prize.

Yet his mathematical interests soon lay elsewhere. He described his first breakthrough in pure mathematics, in his early 20s, as “a nice discovery relating to manifolds and real algebraic varieties”. His peers already recognised the result as an important and remarkable work.

In 1951, Nash left Princeton for MIT. Here, he became interested in the problems of “isometric embedding”, which asks whether it is possible to embed abstractly defined geometries into real-world geometries in such a way that distances are maintained. Nash’s two embedding theorems are considered classics, providing some of the deepest mathematical insights of the last century.

This work on embeddings led him to partial differential equations, which are equations involving flux and rates of change. He devised a way to solve a type of partial differential equation that hitherto had been considered impossible. His technique, later modified by J Moser, is now known as the Nash-Moser theorem.

In the early 1950s, Nash worked during the summers for the RAND Corporation, a civilian thinktank funded by the military in Santa Monica, California. Here, his work on game theory found applications in United States military and diplomatic strategy.

Perhaps Nash’s greatest mathematical work came from studying a mathematical puzzle that had been suggested to him by Louis Nirenberg. It concerned a major open problem concerning elliptic partial differential equations. Within a few months, Nash had solved the problem. It is thought that his work would have won him the Fields Medal — the most prestigious prize in maths, open only to those under 40 — had it not been solved at the same time by Italian mathematician Ennio De Giorgi. The men used different methods, and were not aware of each other’s work — the result is known as the Nash-De Giorgi theorem.

One of the many amazing aspects of Nash’s career was that he was not a specialist. Unlike almost all top mathematicians now, he worked on his own, and relished attacking famous open problems, often coming up with completely new ways of thinking. Louis Nirenberg once said, “About 20 years ago somebody asked me, “Are there any mathematicians you would consider as geniuses?” I said, “I can think of one, and that’s John Nash.” He had a remarkable mind. He thought about things differently from other people.”

In 1959, Nash began to suffer from delusions and extreme paranoia. For the next 40 years or so he was only able to do serious mathematical research in brief periods of lucidity.

Remarkably, however, he gradually improved and his mental state had recovered by the time he won the Nobel in 1994.

Nash showed such resolve and stamina in his mathematical work and in recovering from his mental illness, that his death in a taxi crash on the New Jersey turnpike seems all the more pointless and tragic

May his soul rest in peace.