(Real) As a graduate student in Princeton, Nash (Russel Crowe) attacked several difficult problems. Nash Equilibrium, a concept in game theory won him a Nobel 40 years later. Apart from that Nash also got good results in topology. He discussed his ideas with giants like Einstein, John von Neumann and other faculty members in his math department.
(Movie) Nash is playing a game with his friend and rival Hansen. (Real) While at Princeton, Nash also invented a game called Nash which became quite popular on the campus. After his PhD, Nash went to MIT as a faculty and produced very good results.
(Movie) Nash meets his prodigal roommate Charles Herman (Bettany) in Princeton, a literature student. At MIT, he meets William Parcher (Harris) of US Dept of Defense and starts working for him to crack secret Russian codes transmitted through everyday newspapers / magazines.
(Real) In early 1959, Nash delivered lectures at Columbia and Yale on Riemann Hypothesis. Both talks were incoherent and gibberish. Audience realized that something was wrong. By April 1959, he was admitted to McLean Hospital near Boston. Later, he was also admitted to two other hospitals near Princeton.
(Movie, Real) In mental hospitals, Nash underwent treatments like insulin coma therapy.