Why mathematician John Forbes Nash Jr. will be remembered forever
“What truly is logic? Who decides reason? My quest has taken me to the physical, the metaphysical, the delusional, and back.” Do these lines ring a bell? If yes, then you have definitely watched A Beautiful Mind. A biographical drama based on the distinguished American mathematician John Nash; it explores the life of this mathematical genius. Directed by Ron Howard, the Hollywood movie released in 2001 and stars Russell Crowe in the lead role.
Nash is the one who revolutionised the mathematical model of Game Theory (study of mathematical models of conflict and cooperation among intelligent, logical decision-makers, like us humans) for which he received the 1994 Nobel Prize in Economics. Nash was one of the best mathematical researchers and thinkers of the 20th century. He pursued PhD in Mathematics from Princeton University, USA with a 28-page dissertation on 'non-cooperative games' in 1950.
Born as John Forbes Nash Jr. on June 13, 1928 in New Jersey, USA, his fundamental contribution to the Game Theory was known as the Nash Equilibrium. Here are three of his most invaluable contributions to the field of mathematics.
He gifted us the concept of Nash Equilibrium
In simple words, Nash Equilibrium helps participants of a game to figure out mathematically and logically the actions they should take for the best outcomes. According to this equilibrium, one player’s decision is based on the opponent’s strategy. All the participants will continue with their chosen strategies, as they have no incentive to change or deviate from it. Nash Equilibrium can be utilized in various facets of life.
He developed the isometric embedding
Nash Jr. also worked extensively on differential geometry. It is a branch of mathematics concerned with the geometry of smooth objects and spaces, sometimes known as smooth manifolds. Differential calculus, integral calculus, linear algebra, and multi-linear algebra techniques are used in this field of geometry. Nash Jr. developed the concept of isometric embedding (an embedding that does not modify the surface's inherent geometry), where he placed a surface into space without tearing, creasing, or crossing it.
Nash Jr. observed that some surfaces were so heavily curved or twisted that they couldn't be fully integrated in three-dimensional spaces. They couldn't even be placed in a four-dimensional space. However, with his experiments, he was able to prove that any surface could be embedded in a 17- dimensional space! Extra dimensions, rather than complicating the task, made it easier by allowing him more place to embed the surface. Nash Jr. demonstrated that a manifold could be embedded in space of any dimension without affecting its geometry.
He contributed to partial differential equations
Nash Jr.'s research on isometric embeddings led him to partial differential equations, which are equations that involve flow and rates of change. He found a method for solving a form of partial differential equation that had previously been thought impossible to solve. The Nash-Moser theorem is named for this technique, which was later refined by Jurgen Moser.