The equations seen on the classroom chalk boards are actual equations written by the real-life John Nash.
John Nash visited the set, and Russell Crowe said later that he had been fascinated by the way he moved his hands, and he had tried to do the same thing in the movie. He thought it would help him get into the character.
The film was shot in sequence in order to help Russell Crowe develop a consistently progressive manner of behavior.
Nash's mutterings after he loses the board game (along the lines of "the game is flawed," "I had the first move, I should have won") are in reference to "Game Theory," the economic theory that John Nash is probably most famous for.
John and Alicia Nash were killed in a car accident on May 23, 2015, when the driver of a taxi they were riding in lost control of the car and collided with a guard rail and a second car on the New Jersey Turnpike. The New York Times said that although the two drivers suffered only minor injuries, the Nashes were "ejected from the cab and pronounced dead at the scene." John was 86 and Alicia was 82.
Bryce Dallas Howard: The director's daughter and actress Bryce Dallas Howard appears midway through the movie, after the math lecture when John Nash (Russell Crowe) is sedated by Dr. Rosen (Christopher Plummer), looking into the car as it drives away.