# John F. Nash's Work in Pure Mathematics

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2022-01-14 18:30:03

It’s fair to say that John F. Nash Jr. (1928-2015) didn’t earn much respect from pure mathematicians for his work in game theory. To them, his use of L.E.J. Brouwer (1881-1966) and Shizuo Kakutani (1911-2004)’s fixed-point theorems to show the existence of Nash equilibria in finite games with mixed strategies was and perhaps still is merely considered “not surprising applications of well-known methods” (Milnor, 1998). Indeed, even John von Neumann (1903-1957)—when confronted with Nash’s result—is to have replied “That’s trivial, you know. That’s just a fixed point theorem”.

Although—by far—recognized most widely for the work he did in game theory, Nash in the 1950s however also wrote groundbreaking papers on real algebraic geometry, topology and partial differential equations. Indeed, he won the Abel Prize in 2015 for his work on the latter, alongside Louis Nirenberg (1926-2020), considered one of the most outstanding pure mathematicians of the 20th century. For pure mathematicians, this subsequent work of Nash’s, which was accomplished just before his illness set in, is considered far more rich and important.

A bibliography assembled by his friend John Milnor (1931-) in 1998 lists 21 publications to Nash’s name. However, excluding his work in game theory and economics, as well as conference papers and book chapters, Nash’s original work in pure mathematics is essentially contained in four journal papers, written between 1952-58: