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# A Beautiful Mind: Chapter 20 Summary & Analysis

Summary
Analysis
Warren Ambrose, a colleague of Nash’s at MIT, pens an angry letter to another mathematician in the spring of 1953, accusing Nash of being a “childish bright guy” with few practical skills in mathematics. A “moody, intense, somewhat frustrated” mathematician, Ambrose cultivates an intense rivalry with Nash, who makes him the target of several cruel pranks. Eventually, Ambrose challenges Nash to solve the “embedding problems for manifolds,” a notoriously difficult problem. Nash does so, in part because he is a mathematician who views mathematics as a “collection of challenging problems” to be addressed and solved.
Since Nash treats his encounters with other mathematicians as games to be won, he thrives off of competition and rivalry. When Ambrose challenges him to solve one of the most difficult problems in all of mathematics, Nash gladly rises to the occasion. Nash’s hunger for knowledge and success made him a formidable competitor, and he constantly challenged himself to come up with new ways to solve old (and famously difficult) problems.
Themes
“Embedding” is the process of portraying a geometric object as “some space in some dimension.” Nash’s theorem for embedding states that any kind of manifold embodying “a special notion of smoothness” can be embedded in Euclidean space, a certain kind of two- or three-dimensional space: in other words, “you could fold the manifold like a silk handkerchief, without distorting it.” Nash had to solve a certain set of partial differential equations in order to come up with this result—equations that at the time were impossible to solve with existing methods. Though few believe Nash will solve the problem—even Levinson—Nash is like a “long-distance runner”: he perseveres where others give up and is a “hard worker by habit,” determined to prove that what seems impossible might be possible—in his own hands. In October 1954, Nash’s manuscript on embedding is accepted to the Annals of Mathematics, the prestigious Princeton mathematics journal.
Nash’s genius as a truly innovative thinker lies in his ability to think outside of the box, coming up with ways to attack difficult problems that no other mathematicians would have considered. Nasar describes Nash as a “long-distance runner” to emphasize his unique skills as a scholar. Whereas other mathematicians might focus on fast, flashy puzzles, Nash spent years pondering solutions to seemingly impossible problems.
Themes
Related Quotes
In the early part of 1953, Martin offers Nash a permanent faculty position at MIT, which comes as a surprise to faculty members who believe that Nash is “disdainful” and a bad teacher to boot. Nonetheless, when Nash solves the embedding theorem, many of the faculty members are pleased to support him—including Warren Ambrose, who had been skeptical of Nash’s ability to produce a result.
For all of Nash’s success as an early career mathematician, his difficult personality makes him the object of much criticism, even at his own institution: again, it is clear that Nash’s genius cannot always make up for his faults as an individual.
Themes