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In 1947, John Nash (Russell Crowe) arrives at Princeton University. He is co-recipient, with Martin Hansen (Josh Lucas), of the prestigious Carnegie Scholarship for mathematics. At a reception he meets a group of other promising math and science graduate students, Richard Sol (Adam Goldberg), Ainsley (Jason Gray-Stanford), and Bender (Anthony Rapp). He also meets his roommate Charles Herman (Paul Bettany), a literature student, and an unlikely friendship begins.
Nash comes under increasing pressure to publish, both from the mathematics department chairman and in the form of rivalry with Hansen. But he refuses until he finds a truly original idea. His inspiration comes when he and his fellow graduate students discuss how to approach a group of women at a bar. Hansen quotes Adam Smith and advocates "every man for himself", but Nash argues that a cooperative approach would lead to better chances of success. This leads to a new concept of governing dynamics which Nash develops and publishes. On the strength of this he is offered an appointment at MIT where Sol and Bender join him.
Some years later, Nash is invited to the Pentagon to crack encrypted enemy telecommunication. Nash is able to decipher the code mentally, to the astonishment of other codebreakers. He considers his regular duties at MIT uninteresting and beneath his talents, so he is pleased to be given a new assignment by mysterious supervisor William Parcher (Ed Harris) of the United States Department of Defense, to look for patterns in magazines and newspapers in order to thwart a Soviet plot. Nash becomes increasingly obsessive about searching for these hidden patterns and believes he is followed when he delivers his results to a secret mailbox.
Meanwhile a student, Alicia Larde (Jennifer Connelly), asks him to dinner, and the two fall in love. On a return visit to Princeton, Nash runs into his former roommate Charles and meets Charles' young niece Marcee (Vivien Cardone), whom he adores. With Charles' encouragement he proposes to Alicia and they marry.
Nash begins to fear for his life after witnessing a shootout between Parcher and Soviet agents. He tells Parcher that he wants to quit his special assignment but Parcher blackmails him into staying. While delivering a guest lecture at Harvard University (with Charles and Marcee present), Nash attempts to flee from what appear to be foreign agents, led by Dr. Rosen (Christopher Plummer). After punching Rosen in an attempt to flee, Nash is forcibly sedated and sent to a psychiatric facility. He believes the facility is run by the Soviets who are trying to extract information from him.
Dr. Rosen tells Alicia that Nash has schizophrenia and that Charles, Marcee and Parcher exist only in his imagination. Alicia investigates and finally confronts Nash with the unopened documents he had delivered to the secret mailbox. Nash is given a course of insulin shock therapy and eventually released. Frustrated with the side-effects of the antipsychotic medication he is taking, he secretly stops taking it. But this causes a relapse and he meets Parcher again.
After an incident where Nash endangers his infant son and accidentally knocks Alicia and the baby to the ground (thinking he's stopping Parcher from killing her), she flees the house in fear with their child. Nash steps in front of her car to prevent her from leaving. He tells Alicia, "She never gets old", referring to Marcee, who although years have passed since their first encounter, has remained exactly the same age and is still a little girl. With this, he finally accepts that although all three people seem real, they are in fact part of his hallucinations. Against Dr. Rosen's advice, Nash decides not to restart his medication, believing that he can deal with his symptoms in another way. Alicia decides to stay and support him in this.
Nash approaches his old friend and rival, Martin Hansen, now head of the Princeton mathematics department, who grants him permission to work out of the library and audit classes. Years pass and as Nash grows older he learns to ignore his hallucinations. Eventually he earns the privilege of teaching again.
In 1994, Nash is honored by his fellow professors for his achievement in mathematics, and goes on to win the Nobel Memorial Prize in Economics for his revolutionary work on game theory. The movie ends as Nash and Alicia leave the auditorium in Stockholm; Nash sees Charles, Marcee, and Parcher standing to one side and watching him.