Skip to content

John Nash – Riemann manifolds

  • maths

John Nash, mathematician and Nobel laureate in economics, died in a taxi accident on May 23; at 86. His wife, Alicia, was with him and also did not survive the crash. They are coming home to Princeton from Norway, where John has been honored as the winner (along with Louis Nirenberg) of that year’s Abel Prize in Mathematics. Thanks to A Beautiful Mind, Sylvia Nassar’s chronicle of Nash’s life, and its film adaptation starring Russell Crowe, Nash was one of the few mathematicians well known outside the halls of academia. The general public may remember the story of Nash’s mental illness and his recovery from paranoid schizophrenia.

But Nash’s influence goes beyond the Hollywood version of his biography. His colleagues consider his mathematical innovations, especially in non-cooperative games (the work that would win him the Nobel Prize), among the great economic ideas of the 20th century.
Nash is best known for his work in game theory. In mathematics, a ‘game’ involves two or more ‘players’ who earn rewards or punishments depending on the actions of all participants. Some are called zero-sum games, meaning that one player’s gain is another player’s loss. Nash’s work applies to non-cooperative games. In these situations, players can unilaterally change their strategy to improve (or worsen) their own score without affecting other players.

The prototypical example of such a game is the basic ‘prisoner’s dilemma’. Two criminals are captured and kept in separate cells, unable to communicate with each other. Prosecutors do not have enough evidence to convict them on the main charge, but they can convict them on a lesser charge that comes with a one-year sentence. The prisoners are offered a deal: testify against the other defendant (i.e. turn him in) and be released as long as he serves three years. However, if both defendants surrender, both will serve two years. If neither betrays the other (i.e. cooperates), then both will be convicted of the lesser charge and serve one year. The results can be summarized in a matrix.
What Nash discovered is that every such game has a strategy, now called a Nash equilibrium, where any unilateral change in strategy by a player results in a worse outcome for that player. In the case of the prisoner’s dilemma, there are two such equilibria, the upper left and lower right squares of the matrix. Indeed, in the lower-right situation, if one of the players changes his strategy unilaterally and decides not to defect, he will increase his penalty and thus end up with a worse outcome. This example is particularly unpleasant because the upper left strategy is clearly the best way for the prisoners (they must remain silent), but purely rational players will end up in the lower right position. Game theory has applications in many fields, including economics and political science.
Many scenarios in international relations can be modeled as non-cooperative games. For example, the development of nuclear programs during World War II can be modeled as a kind of prisoner’s dilemma in which both sides decide to develop the atomic bomb out of fear that the other side will. This, of course, led to the less desirable outcome of nuclear proliferation, analogous to the defection of the two prisoners.

Although Nash is best known worldwide for his work in game theory, most mathematicians consider his results on embeddings of Rieman manifolds to be his most innovative and important. In this subspecialty of geometry, an n-manifold is a space that locally looks like n-dimensional Euclidean space (the typical three spatial dimensions we’ve used to form 3-dimensional Euclidean space). For example, a surface such as a sphere or a hollow donut is a 2-manifold because every point on the surface has a small disk around it; to a small bug standing at the point, the surface appears as a flat two-dimensional plane (hence the ancient belief that the Earth is flat).

A manifold is Riemannian if there is a globally consistent way to define angles between vectors tangent to the manifold at a point. In particular, it allows us to define distances between manifold points and find the lengths of curves embedded in the manifold. Euclidean space with its usual concept of angle and distance is the simplest example. Now imagine trying to fit an abstract Riemannian manifold into Euclidean space. You can distort it and do all sorts of weird things that end up distorting the angles between the tangent vectors of your manifold.

The Nash-Kuiper embedding theorem states that we can always correct this problem; that is, we can find a realization of a Riemannian manifold of dimension n in Euclidean space of dimension n+1 such that the angles are conserved. You can then more easily calculate distances between points on the manifold using a Riemannian structure inherited from Euclidean space. This may not sound earth-shattering, but the problem has plagued mathematicians for more than a century. That the size of Euclidean space cannot be made smaller than n+1 is familiar to anyone who has studied a map – the surface of a sphere cannot be flattened into a plane without distorting the corners. There are many counterintuitive consequences of Nash’s theorem.
For example, it implies that any closed surface can be realized inside an arbitrarily small ball in 3-dimensional space.
Nash is also credited with inventing a game eventually marketed by Parker Brothers as a board game called Hex. This game, played on a parallelogram-shaped field of hexagonal cells, was discovered independently in Denmark around the same time. At Princeton, it’s called Nash, after its creator, or John, a double entendre involving the fact that it’s played on the tiles in the men’s room in the math department. There are two players, each with tokens of one color (red and blue, say). The goal is to form a continuous path from one side of the board to the other before the opponent does the same in the opposite direction. There are online versions of the game. The first player always has a winning strategy; this means that the player who makes the first move can always win,

In every single century, there are a handful of mathematicians who stand out, whose work is so original and innovative that it becomes part of the language. As journalist Erika Klareich pointed out, no one cites Nash’s papers anymore because “Nash equilibrium” is standard vocabulary; every mathematician knows what it means. Although he published only a small handful of papers, John Nash will be remembered as one of the most original and influential mathematicians of the 20th century, whose work continues to inspire new results and research directions.