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In Chicago, on the day of her twenty-seventh birthday, Catherine (Gwyneth Paltrow) receives her sister Claire (Hope Davis) from New York for the funeral of their father Robert (Anthony Hopkins), who passed away a couple of days before. Robert was a brilliant mathematician that became bughouse, and Catherine had lived with him for the last five years, inclusive quitting her studies in the faculty, and she is concerned about having inherited his insanity. The mathematician of the University of Chicago, Hal (Jake Gyllenhaal), is making a research in Robert's notebooks, trying to find any brilliant proof that Robert might have produced in one moment of lucidity. When Hal has one nightstand with Catherine, she gives a notebook to him with the development of a unique mathematics theory that Catherine claims that she developed. Hal and Claire do not believe on her, until the truth is disclosed.