The film begins with a poignant conversation between Catherine (Gwyneth Paltrow) and her father Robert (Anthony Hopkins) during a late-night moment where he surprises her while she's watching TV. He gifts her a bottle of champagne for her birthday, which she humorously critiques for its poor taste. As they discuss the complexities of insanity, Catherine challenges Robert's assertion of his own madness by pointing out that true insanity prevents one from acknowledging their condition. In a twist, Robert reminds her he isn't merely crazy; in fact, he passed away last week, with his funeral set for the following day.
Catherine awakens from this disorienting dream to discover that Hal (Jake Gyllenhaal), a former student of Robert's, is still in the house. Hal is an aspiring mathematician attempting to carve a niche for himself in a field dominated by giants like Robert, who made groundbreaking contributions by the age of 22 but battled severe mental illness for over two decades. Hal admits he believes there may be undiscovered notes of Robert’s genius lurking among his texts. While Catherine initially interrogates Hal, suspecting ulterior motives, he insists he would honor Robert's legacy in any discoveries he makes.
Their exchange escalates as Catherine searches through Hal's backpack for her father's belongings, eventually resulting in a notebook falling out, which Hal intended to give her as a gift, containing a compliment about her. After a quick call to the police, she realizes they need not intervene.
The following day, as the family gathers for Robert's funeral, Catherine's sister Claire (Hope Davis) arrives in a lavish town car, highlighting the stark contrast between her polished demeanor and Catherine's disheveled appearance. Their strained relationship is evident as Claire criticizes Catherine’s appearance and their differing approaches to family obligations come to the forefront. Flashbacks reveal Catherine’s difficulty balancing her graduate studies with the looming responsibility of caring for her increasingly dependent father.
During the funeral service held at the impressive Rockefeller Chapel, Catherine's frustration over the many attendees—none of whom supported Robert in his final years—boils over as she publicly vents her anger, even stating her relief over his passing. After her dramatic exit, Hal offers to walk her home, which she sharply declines.
Amidst Claire's plans to sell Robert's house back to the University, Catherine grapples with losing her home and her fears of potentially inheriting her father's mental health struggles. The night of the wake, while Claire indulges in drinks with an academic, Hal and Catherine find solace in each other, leading to an intimate encounter.
Post-encounter, Catherine grows increasingly comfortable with Hal, whom she grants access to her father's desk, where he uncovers a potentially monumental mathematical proof. Thrilled about this discovery, he presents it to Catherine and Claire, only to be met with disbelief. A pivotal confrontation unfolds as Catherine asserts authorship, claiming the proof is hers, causing Hal and Claire to doubt her capabilities. Claire’s insecurities lead to an intense confrontation, raising questions about Catherine's mental state.
The next day, as tensions escalate, Catherine experiences a breakdown, revealing her internal struggles and guilt, voicing, “I stole it from him.” Claire reassures her, emphasizing the extreme stress they both endured during their father's illness.
Hal then takes the proof to his colleagues for validation, who deem it credible. Meanwhile, Claire's influence leads Catherine towards a departure to New York, but Hal, desperate to connect with her, encourages her to recognize her worth. As she contemplates her identity and her father’s legacy, a poignant flashback illustrates her challenging journey to mastery in mathematics under Robert’s guidance, only for her to confront her father in a state of confusion from his illness.
In a reflective moment at the airport, Catherine gradually acknowledges her own abilities and resolves that she need not follow her sister. Racing against time, Catherine rushes back to the University of Chicago, leading to a heartfelt reunion with Hal on campus, where they engage in a much-awaited discussion about the proof, symbolizing her acceptance of her genius as students bustle around them.
