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