Proof (PG-13, 2005)

Following the death of her father, Catherine (Gwyneth Paltrow) begins to worry that she might be insane. Her father, renowned and brilliant mathematician Robert (Anthony Hopkins), for a long time suffered from mental illness and Catherine had dropped out of school to care for her him. Following the funeral, one of Robert's former students, Hal (Jake Gyllenhaal), finds a proof that he believes to have been Robert's, yet Catherine claims it was hers. While Hal seeks to validate the authenticity of the proof, Claire (Hope Davis), Catherine's sister, convinces Catherine to come with her to New York.

Based on David Auburn's long-running play (on Broadway and on London's West End, among other venues), John Madden's PROOF poses questions of trust and doubt, grief and guilt, ambition and selfishness, as well as the sisters' competition and resentment. Catherine is also presently dealing with Hal (Jake Gyllenhaal), an aspiring mathematician and drummer in a local band, who comes by to sort through Robert's papers, hoping, perhaps, to find a lost instance of genius, something recorded during a rare lucid moment, a last sign that his madness was not utter and all-consuming, even as it may have seemed that way.

The film's central, concrete problem is the revelation of the proof, not quite elegant but exceptional and potentially math-world-changing (again), hidden away inside a locked drawer in Robert's home office. Flashbacks reveal that he and Catherine spent some time working on a problem together, separately, but at the same time, each writing out pages of proof, working late into nights and bent over desks and tables in deep concentration. As their handwriting is similar, it's unclear whether the newly discovered proof is Robert's or Catherine's. She claims it is hers, but neither Hal nor Claire quite believes it. And so the film presents a knot of questions, twisted up inside a knot of delicate performances and a fragmented narrative.