I spent a good part of my week discovering the work of Maryam Mirzakhani thanks to an excellent documentary on ARTE. In 2014, this brilliant mathematician became the first Iranian to be honored with the Fields medal, and the first of only two women to date! 👩🏻‍🏫🥇

While I did take several advanced maths courses for my Computer Engineering diploma, I do not have the required level to understand the details of her work on hyperbolic geometry. However, what struck me the most was her approach to problem-solving, something that most engineers learn to do with time and experience.

Maryam’s work on Ratner’s theorem

Her most cited work stems from a simple question: “If we hit a ball on a billiard table, what are the possible ways it could travel?” 🎱

We can picture infinite trajectories in all different directions within the table’s boundaries, but uniting these trajectories in one formula proves to be a far too complicated problem to solve. However, imagine we put a mirror along every side of the billiard table. When the ball hits a side, it’s going to bounce off. But its reflection in the mirror will continue in a straight line: basic rule of axial symmetry! 🪞

Maryam chose to study the straight trajectory of the ball’s reflection in an infinity of mirrors, which linked back to some of her previous work on moduli spaces.

Simply put, she transformed a complex problem in a simple setting (a regular billiard table), into a simple problem (straight trajectory) in a complex setting! 📈

Approaching problem-solving

Her problem-solving approach made me think of all current advances in Machine Learning thanks to LLMs, where algorithms looking to make sense of natural language have grown in complexity, yet the data itself is kept simple. A few years ago, researchers were finding it hard to organize data in vast and complex knowledge bases, so that querying information becomes simpler. Two different approaches to tackling similar use cases; yet, one is proving to be much more effective. ✨

Yet another reminder that we should always think outside of the box whenever we hit a wall! 📦

“How can we approach our problem differently?”

“Is there any other way to solve this issue?”

Watch the documentary about Maryam Mirzakhani’s work on YouTube.