At 17, Hannah Cairo Solved a Major Math Mystery

For Quanta Magazine, Kevin Hartnett profiles Hannah Cairo, a math prodigy who disproved the Mizohata-Takeuchi conjecture—a problem mathematicians have been trying to solve for 40 years. During the pandemic, Cairo, who had been homeschooled, immersed herself deeper into mathematics, discovering a world of endless thinking. “Math became a kind of escape, a space that felt expansive when her daily life was not,” writes Hartnett. After completing the proof, Cairo decided to skip completing high school and attending college—and is instead about to start a doctoral program this fall. Hartnett writes an inspiring story about an exceptional teen.

“I went to office hours and asked him, ‘Do these ideas work?’ It turned out they didn’t because they were silly,” she said. “There’d be this back-and-forth. I’d come to office hours with new ideas and ask if they work. And he’d say no.”

Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere.

But Cairo showed that in her function, they didn’t cancel out as expected. Instead, their interference created uneven patterns, causing the function’s energy to spread out over some areas and concentrate in others in a fractal-like way that the Mizohata-Takeuchi conjecture prohibited. She found herself staring at a mathematical construction that by many accounts shouldn’t exist.