## Leonhard Euler

### Good Will Hunting

Sanem Naz Kafalı

For this month’s issue of the Math Chronicles, the movie we have chosen is "Good Will Hunting", which is a 1997 film written by Matt Damon and Ben Affleck, and directed by Gus Van Sant. The film is about a 21-year-old self-taught math prodigy from South Boston, who has just been released from prison. He works as a janitor at MIT. He struggles to comprehend his own troubled past and to find where he fits in the world. He spends the majority of his time drinking with Morgan, Chuckie, and Billy.

Professor Gerald Lambeau, a Fields winner mathematics professor at MIT, gives his students a challenging mathematics problem to tackle. He writes the problem on the blackboard not waiting for an answer for at least a while. Will solves it secretly, surprising both Professor Gerald Lambeau and his students. Professor Gerald Lambeau refers to this problem as a “Fourier equation” despite the fact that it involves graph theory.

Will creates the following adjacency matrix to solve the first part of the problem.

He attempts to locate the third power matrix by counting the number of 3-step walks that are present in the graph.

The third and fourth parts of the question are about generating operations (Wikipedia).

The stunned professor decides to challenge this unknown genius with an even a harder problem, which took almost 2 years for him to prove. The question asks to state Cayley's formula and to draw all the homomorphically irreducible trees with n= 10. Will attempts to write the trees on the board but gets interrupted by the professor who thinks that Will is drawing on his student’s homework. When he realizes the trees on the board are right, he decides to find this brilliant young boy and help him.

In this heartwarming tale about friendship, and self-discovery Will displays his exceptional mathematical skills. He effortlessly solves many more mathematical puzzles like the ones on the blackboard. He finds solutions for problems that even the Fields winner Lambeau can’t find a solution for. One of the most prominent of these is the "Schwarzschild metric" which defines the behavior of black holes in space.

Throughout the film Will not only starts to heal from his past trauma but also makes valuable a friendship with his therapist psychiatrist Dr. Sean Maguireto and starts dating an incredibly smart lady who goes to Harvard named Skylar.

Ultimately, "Good Will Hunting" is a beautifully made movie that anyone who enjoys comedy, romance, or drama at least a bit will enjoy. This award-winning film is definitely something anyone who loves maths must watch.

References

“Good Will Hunting.” Wikipedia, Wikimedia Foundation, 21 Jan. 2023, https://en.wikipedia.org/wiki/Good_Will_Hunting.