The Shape of Things and the 2015 Abel Prize

Indexby Jonathan Kujawa

In Oslo on May 19 John Nash and Louis Nirenberg received the 2015 Abel Prize “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis”. The Abel Prize is barely a decade old but has quickly became one of the most prestigious awards in mathematics. To learn more about this year's winners, visit the Abel Prize webpage here. For an insight into the personalities of the two winners, I especially recommend these short videos.

This year's prize comes with sad news. On their way home from the award ceremony, John and Alicia Nash were killed in an auto accident. You can read the New York Times obituary here.

Last year at 3QD we talked about Yakov Sinai's work in dynamical systems. By coincidence this year's winners' work is closely related to the “exotic” non-Euclidean geometries we discussed at 3QD in March. It's a good chance to dig a little deeper into these topics and get the flavor of Nash and Nuremberg's work. Like last year I should say straight off that I'm not an expert, but I'm happy to talk about some cool mathematics.

John Nash, of course, is one of the most widely known mathematicians of the twentieth century. His life story was told by Sylvia Nasar in “A Beautiful Mind”. The book was made into an award-winning film of the same name starring Russell Crowe. It tells of Nash's brilliant work as a young man and his subsequent difficulties with mental health issues. It's a dramatic story and well worth watching the film. It should go without saying, but the movie turns the drama knob up to eleven and shouldn't be taken as an accurate depiction of Nash's life. For a more nuanced version of events I recommend Nasar's book.

The movie closes with John Nash winning the Nobel prize in Economics for his work in game theory. In game theory we use mathematics to study potential strategies, outcomes, etc., when two or more players are in competition. If you only think about tic-tac-toe, chess, and other such games it first it sounds like a mathematical trifle. But once you begin to look around you see players in competition everywhere: people and corporations in the marketplace, countries in geopolitics, species in evolutionary competition, etc. Game theory is serious business!

In his remarkably short (27 pages!) PhD thesis, Nash studied the following situation. Imagine there is a competition with some number of players, each with their own collection of strategies which they can use as they choose, and a “payoff function” which scores the outcome given each player's choice of strategy. Many real world situations can be fit into such a scenario.

One thing to keep in mind, though, is that there is no room for chance in this sort of game: each player picks their strategy and the payoff function scores how well you and everyone else did in light of everyone's choices of strategy. You would think, then, that if everyone were to lay their cards on the table, so to speak, by revealing which strategy they planned to use, then at least one player should always be able use this knowledge to improve their outcome by tweaking their strategy.

Amazingly, Nash proved exactly the opposite. He proved that given any such competition — no matter how many players, what sorts of strategies they can employ, and how the payoff function scores the players — there will always be at least one configuration of strategies in which no single player can improve his outcome. Such a situation is now called a Nash Equilibrium of the competition.

Now it could be that if the players collaborate they may be able to collectively find a new configuration of strategies which makes some players better off. But in a pure competition where each player only looks out for their own interests, Nash tells us the players could find themselves trapped in a sub-optimal situation. You don't have to be much of cynic to see Nash Equilibria in the real world. We all might be better off if we payed more in taxes, had smaller militaries, or provided online citizens with strong privacy controls. Instead we each try to pay as little tax as we can, have arms races, and an Internet where you are the product, not the customer. Collectively we could improve our situation but change by a single individual leaves them worse off: paying more in taxes without the concomitant improvement in services, an undefended country in a hostile world, or a cave-dwelling anti-technology Luddite. Nash tells us that this can happen in any competition. Keep Nash in mind the next time some free marketeer tries to convince you that the Market always gets it right.


Image from Geek and Poke [1].

In the New York Times obituary Roger Myerson, an economist at the University of Chicago, compared the impact of Nash equilibrium on economics “to that of the discovery of the DNA double helix in the biological sciences.” But you may have noticed that the Abel Prize announcement made no mention of game theory. Instead Nash and Nirenberg are recognized for their contributions to differential equations and geometry. Indeed, previous Abel Prize winner Mikhail Gromov is quoted in the Abel Prize biography of Nash: “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.”

What then about geometry? As we talked about in March, in addition to the usual planer geometry of Euclid we also have the non-Euclidean spherical and hyperbolic geometries. The essential difference is in how your geometric world bends: Euclidian geometry is in the flat, zero curvature world of a tabletop, spherical geometry is on the positive curvature surface of a sphere, and hyperbolic geometry is in the negative curvature surface of a Pringles potato chip. That is, if you are an ant standing on a tabletop, your world extends flatly in every direction, while on a sphere it bends downwards, and on the chip it bends upward.


Image from [2].

Of course if you have a God's eye view from the outside, it's easy to see which way a shape bends. But if you aren't an astronaut can you still tell that you live on a sphere? Sure, you say, I can just observe how ships sink below the horizon as they sail away. But what if you were a myopic ant? Can you tell the shape of your world just looking in your immediate neighborhood?

In his investigations of these new geometries Gauss discovered his Theorema Egregium (“Remarkable Theorem”) [3]. He proved that curvature is an intrinsic, local feature of a geometric object. That is, at every point you can determine which way the shape bends just from data in your immediate neighborhood. No need to see the shape as a whole, or to see it from a distance like an astronaut.

Gauss's theorem has a number of remarkable consequences. We all know the famous story of Eratosthenes using the solstice sun in Alexandria to measure the circumference of the Earth. But since the curvature of a sphere is just the reciprocal of the radius, even a myopic ant with a working knowledge of differential geometry could use what little he can see to recover Eratosthenes's result. There is a nice article in Wired about the Theorema Egregium and how it explains why corrugating strengthens cardboard and bending a pizza slice keeps it from being floppy.

Gauss's theorem raised an important distinction for geometers. On the one hand you have features of the shape which are intrinsic. These are those which can be measured while living on the shape and don't depend on the ambient space or on the observer having a God's eye view. Those which depend on the ambient space are called extrinsic. As you can imagine, geometers prefer intrinsic features. They consider them to be the “real” qualities of a geometric shape. You can talk about the intrinsic features completely independent of any reference to the ambient space. Extrinsic features feel arbitrary and accidental. They could easily be different the next time you run across a supposedly identical shape.

This raises the question of whether the intrinsic geometry actually determines a shape. In 1916 Weyl considered this question: if you have a putative geometric shape which at every point has positive (sphere-like) curvature, can you realize it as an actual shape in 3-dimensional space? The other winner of the Abel Prize, Nirenberg, proved in his PhD thesis that the answer is yes, you can realize every sphere-like shape as an honest to goodness shape in 3-dimensional space. In other words, by measuring the curvature at every point our myopic ant can reconstruct the shape of his entire world.

John Nash took this question to the next level. There are three kinds of 2-dimensional geometries (Euclidean, spherical, and hyperbolic), but in three and higher dimensions the number of possible geometries explode. These higher dimensional analogues are known as Riemann manifolds. In 2-dimensions we were interested in understanding the geometries where the nearby neighborhood of each point looks like a sheet of paper with some amount of curvature. Riemannian manifolds are spaces where every neighborhood looks like Euclidean space along with some intrinsic higher dimensional version of curvature. The question is then whether every such Riemannian manifold can be constructed in a nice Euclidean ambient space. Nash proved that they can!

This may sound a bit esoteric, but remember that string theory requires a geometry with something like eleven dimensions. There are trillions and trillions of such geometries. They have their own intrinsic features and it is they, not Euclidean geometry, which is the world we may live in. Nevertheless, we are most comfortable when we study old fashioned Euclidean geometry. When we study these exotic string theory geometries it is immensely helpful to know that we can safely pretend they live in Euclidean space. Similarly, Einstein's theory of relativity tells us that spacetime itself is an intrinsically curved 4-dimensional geometry. Nash's theorem tells us we can image spacetime living in a higher dimensional Euclidean space. This may sound like a complicating step, but in fact it's again a big simplification.

Nirenberg and Nash were also recognized for their work in differential equations. Very roughly speaking, this is the area of mathematics which focuses on the following question: if you have some unknown quantity or quantities and you know how they change, can you determine the quantities themselves? This question is ubiquitous in science: you may know the rate of change of a chemical reaction, or how an oil flow in a pipeline changes over time, or how wind velocities change as they flow over a wing in a wind tunnel, but can you describe the reaction/flow/wind-stream itself? Perhaps more surprisingly, differential equations are at the heart of the geometric questions we were just discussing. Nash and, especially, Nirenberg have made fundamental contributions to this area.

One of the most famous open questions in mathematics is about the Navier-Stokes differential equation. In fact, one of the seven $1,000,000 Millennium Prize problems is about solving the Navier-Stokes equation. The Millennium Prize problem is very much in the same spirit as the geometric questions we talked about above. The Navier-Stokes equation describes how fluids flow. The question is if you give me the Navier-Stokes equation for a possible fluid flow, can I find an actual fluid flow in 3-dimensional space which embodies this Navier-Stokes equation? Some of the best work on the Navier-Stokes equation is by Nirenberg working with Cafarelli and Kohn.


Image from [4].

Once again the Abel Prize committee has identified fantastic mathematicians who have done marvelous work. I look forward to next year's winners!

[1] Geek and Poke.

[3] From NASA's website where they discuss the shape of the universe.

[2] Remember, Gauss is also known for the Eureka Theorem. He seems to have had a talent for PR.

[4] Dhiren Mistry: Evolving Swirls, University of Cambridge.

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