A 17-year-old girl refutes a mathematical conjecture proposed 40 years ago.

Hannah Cairo was stuck on a math problem. All she could think about was a few weeks and decided to try a new approach. “After months of trying to prove the result, I managed to understand why it was so difficult. I realized that if I used that information correctly, I might be able to refute the claim. Finally, after several failed attempts, I found a way to construct a counterexample [a case that doesn't verify the property being studied and that shows it isn't generally true].” Ciaro says it required several tools, including fractals, and she had to arrange everything very carefully. “It took me a while to convince Ruixiang Zhang [the professor of the subject in which the problem had been posed] that my proposal was actually correct,” Cairo says.

It turned out to be true, and with that, Cairo solved the so-called Mizohata-Takeuchi conjecture , a problem proposed in the 1980s that the harmonic analysis community had been working on for decades. Although it was widely expected to be true—if so, other important results in the field would automatically be resolved—the community greeted the result with enthusiasm. And with surprise: its author was a 17-year-old girl who hadn't yet graduated from high school.

“When I moved to the US from Nassau [Bahamas, where he was born], I entered the educational system as a high school student, although I took classes at UC Berkeley. I wrote to professors, telling them what books I'd read on the subject, and asking if I could attend their classes. Many said yes, including Zhang,” he says. “One day, he proposed proving a special, much simpler case of the conjecture as a homework assignment. As an optional part, he posed the original conjecture. And I became obsessed with it,” he adds.

The Mizohata-Takeuchi conjecture falls within the field of harmonic analysis, which attempts to decompose functions into simpler components, such as sinusoidal functions. Today, it is a very hot area of ​​research and has also become a fundamental tool in numerous applications, from the compression of digital audio and video files to the design of telecommunications systems.

Harmonic analysis was born in the early 19th century with the work of French mathematician Joseph Fourier on the study of the heat function, a partial differential equation that describes the diffusion of heat in a solid body. His revolutionary idea was to decompose this complex function as a sum of sines and cosines. The technique, known as Fourier series, opened the door to a new way of understanding physical and mathematical phenomena. “In the theory of harmonic analysis, everything is made up of waves. You can build anything with them, if you use the right number of waves,” Cairo describes.

Restricted Fourier analysis examines what kinds of objects can be obtained if we start with just a small set of waves. “Only certain things can be constructed, and it's very difficult to understand which ones. The Mizohata-Takeuchi conjecture states that if you use only certain types of waves, you get a shape made of lines,” he explains.

“Once I obtained the first counterexample, I tried to reformulate the entire problem in frequency space. And I observed what my construction looked like, in this way. Then I realized that, in reality, there was another, much simpler way to design a counterexample,” she declares, satisfied, in one of the rooms of the San José Residence in El Escorial, where the 12th International Congress on Harmonic Analysis and Partial Differential Equations was held from June 9 to 13, organized by the Institute of Mathematical Sciences (ICMAT) and the Autonomous University of Madrid. The event, known as the El Escorial Meetings , has become, in its almost 50-year history, one of the most prestigious in the field.

This is Cairo's first international scientific trip. She landed in Barcelona two weeks ago, and this is her fourth conference since then. "It's a wonderful experience spending time with other people who love mathematics," she says. At the conference in El Escorial, she gave one of the program's talks. And, far from feeling self-conscious, she enjoyed it. Cairo enjoys public speaking. She loves teaching other students—sometimes older than her. Her vocation, she says, is "to help other people, to make them happy." And, for as long as she can remember, she's been fascinated by mathematics.

She began reading complicated textbooks on the subject on her own. “I always wanted to be a mathematician, but I didn't really know what it meant until I learned abstract algebra from books. It's funny, because abstract algebra is the opposite of the mathematics I do now. Actually, at first, I thought I would do number theory. When I was thirteen or fourteen, I wrote a paper on number theory, but it dealt with a problem that nobody cared about,” she recalls, laughing.

During the COVID-19 pandemic, the Berkeley Math Circle summer camp—a gathering where pre-university students collaboratively solve difficult math problems, similar to ICMAT's Small Institute of Mathematics (PIM) —had to be held online. This allowed Cairo, from the Bahamas, to enroll in the course. “Math circles are about exploring and sharing your ideas with friends; they're not like school math, where you have to memorize. The work is similar to painting a picture with your ideas. It's not about achieving a tangible goal, but simply about understanding things, asking questions, and it's also a great way to make friends,” she describes.

The program director recognized Cairo's extraordinary mathematical talent—another goal of this type of activity is to identify people with a special talent for mathematics and foster their interest and ability—and suggested she become a professor in subsequent editions. She did so. Now, at her new university, Maryland, where she will begin her doctorate next year, she hopes to start her own group.

There, she will continue working, supervised by Zhang. “He helped me a lot, and I am very grateful. Beyond his class, which I loved, he dedicated countless hours of tutoring to me,” she recalls. In Spain, ICMAT's new Mathematics Intensive Program (MIP) also seeks to identify and support careers of this kind.

Ágata Timón García-Longoria is the coordinator of the Mathematical Culture Unit at ICMAT.

Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT). In this section, researchers and members of the center describe the latest advances in this discipline, share common ground between mathematics and other social and cultural expressions, and remember those who shaped its development and knew how to transform coffee into theorems. The name evokes the definition of Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems."