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# A Beautiful Mind: Chapter 15 Summary & Analysis

Summary
Analysis
During the 1950s, Nash’s work on game theory (a kind of “applied” mathematics) is not considered important enough to earn him a position as a professor at a top academic department. As a result, Nash begins working on a paper that he hopes will connect his work to “pure” mathematics, focused on geometric objects called manifolds. Nash develops a theorem related to manifolds, deriving the solution before he works out the steps of the proof—just as he had with the bargaining problem. Donald Spencer, a visiting Princeton professor, helps Nash to finetune the theorem, which connects the object of the manifold to a much simpler class of objects, called real algebraic varieties. By finding a link between these kinds of objects, Nash “opened up new avenues for solving problems” in topology, the study of geometric properties.
Though Nash is disappointed that his work in game theory will not help him to become a professor at a top university, he is not daunted. Characteristically stubborn and determined to achieve success at all costs, Nash decides to shift his focus to “pure” mathematics: his skills as a brilliant thinker help him to make important discoveries in this field, as well as in applied mathematics.
Themes
Though Nash’s paper on manifolds helps to establish him as a “pure mathematician of the first rank,” it does not help him to obtain an assistant professorship at Princeton. Emil Artin voices his opposition to appointing Nash as a professor, viewing Nash as “aggressive, abrasive, and arrogant.” However, MIT (Massachusetts Institute of Technology) offers Nash an instructorship, which Nash accepts.
Despite his genius, Nash’s abrasive behavior makes him a less-than-ideal candidate for a professorship at Princeton, suggesting his achievements and intelligence cannot make up for his significant personal failings.
Themes