Cairo followed this advice. In the fall of 2023, her family moved to Davis, 60 miles northeast of Berkeley. There, her older brother enrolled as a freshman at UC Davis, and her parents allowed her to commute to Berkeley on Tuesdays and Thursdays. By the spring, she was going five days a week and taking several more classes. She recalls it as a time in her life when she began to feel full of possibility.
“I had started making friends, and I was feeling good,” she said. After the spring semester ended, her family moved from Davis to Berkeley — her brother had decided to transfer there — and Cairo finally felt able to settle in.
Even so, it was an adjustment. “I didn’t have many social experiences, so I still had to learn how to interact with other humans,” she said.
As the 2024–2025 academic year neared, Cairo considered what courses she would take. One class in particular caught her eye — a graduate course in Fourier restriction theory, a branch of harmonic analysis. “It was one of the most advanced analysis classes being offered that semester, so I thought, I’ll just go take it,” she said.
The mathematician Ruixiang Zhang gave Cairo a homework problem that she couldn’t stop thinking about. He later became her adviser. Lin Lin
The professor for the course was Ruixiang Zhang, an accomplished mathematician whose path into the field had followed a more traditional arc: a gold medal at the 2008 International Mathematical Olympiad, the prestigious high school competition; a doctorate from Princeton University; a postdoc at the Institute for Advanced Study; a tenure-track position at Berkeley, one of the top math departments in the world.
Cairo emailed Zhang, asking to enroll. “Hannah was very focused and seemed to be passionate about the topic,” he said. “This attitude alone is enough for me, so I just gave her permission.”
Within a few weeks, while working on a problem set, she came across a problem that she couldn’t stop thinking about.
Extra Credit
The problem was a simplified version of the Mizohata-Takeuchi conjecture. Zhang had included it in one of his homework assignments as a warm-up, hoping to encourage students to practice advanced techniques in a deep area of math. The assignment also included an optional extension, inviting them to consider whether the proof they’d found for the simplified case could be extended to more complicated formulations of the problem.
Cairo completed the problem set and took Zhang up on the invitation to keep thinking. To her, it seemed natural to follow the thread of an idea as far as it would go. “Why would I stop?” she said.
The Mizohata-Takeuchi conjecture is a problem in harmonic analysis, a field that studies how functions are assembled from wavelike components.
Any given function can be written as the sum of simpler wavelike pieces, called sine waves. Each of those sine waves, in turn, has a frequency. Mathematicians often want to understand the nature of functions that can only be built out of sine waves with certain frequencies. In these cases, the only permitted frequencies are those that satisfy equations that carve out specific surfaces, like a sphere. That’s because the functions that define many physical waves — such as light, sound and quantum particles — are restricted to these kinds of frequencies.