17-year-old girl solves 40-year-old math mystery

In February, the international mathematics community was shocked when a scientific paper refuted a famous conjecture that had seemed true for more than four decades.

The author is not a professor or a veteran researcher, but Hannah Cairo - a 17-year-old female student.

The conjecture Cairo rejected was called the Mizohata-Takeuchi conjecture, and it concerned the way functions are formed from sine waves—the foundation of harmonic analysis. For years, mathematicians had been trying to prove the conjecture correct. But Cairo found a counterargument: it was wrong. The discovery caused an immediate sensation, and at the same time brought the name of an extraordinary young talent to the public.

From homeschooling to graduate school

Born in the Bahamas, Hannah was homeschooled with her two brothers. By age 11, she had completed her calculus coursework online and had mastered advanced university courses on her own. But homeschooling felt confining and lonely. For Hannah, mathematics became a wide open world to explore at any time, using only her imagination.

Hannah Cairo believes that "Mathematics is an art form". (Photo: Scientific American)

The turning point came when the COVID-19 pandemic left the family stranded in the US. This gave Hannah the opportunity to join math clubs and an online summer camp at the University of California, Berkeley. Her extensive self-study resume impressed faculty, paving the way for Hannah to enroll in Berkeley’s dual-degree program, where she began taking graduate math classes, despite still not having a high school diploma.

In one assignment, Professor Ruixiang Zhang—a 2008 International Mathematical Olympiad gold medalist and PhD holder from Princeton—presented a simplified version of the Mizohata-Takeuchi conjecture for students to practice. Hannah didn’t stop there: She dug deeper, tested more ideas, and eventually found a counter-argument that brought the original conjecture to its knees. This discovery not only changed the landscape of a field, but also led to the collapse of many related conjectures.

Even she was skeptical at first. “I often think I have a proof, only to find out it’s wrong,” Cairo said. But this time, two things were different: She realized she could replace it with a simpler structure that would still give the same result, and, more important, she convinced both herself and Professor Zhang that the proof was correct.

Break all limits

Cairo’s journey to conquer the difficult problem was also challenging. Due to her lack of experience, her initial efforts were often flawed. “Every week I would bring a new idea to his office and ask, ‘Is this right?’ And he would say no,” Cairo recalls.

Hannah Cairo says she enjoys presenting math with colorful, cute slices. (Photo: Quanta Magazine)

But instead of getting discouraged, she continued reading, thinking, and experimenting. Eventually, Cairo constructed a strange function from the waves of frequency on the curved surface—just as the hypothesis required. Instead of canceling and amplifying each other as they normally would, the waves scattered their energy in irregular, almost fractal-like patterns—something the hypothesis claimed should not happen.

“Cairo’s paper shows that seemingly obvious and elegant hypotheses can still fall apart in unexpected ways,” says mathematician Fernando Oliveira. “But to see it, you need the right ‘scope.’”

Mathematician Anthony Carbery said: “I was like, ‘Wow!’ This is a problem I have loved for 40 years. Cairo’s solution is not only fascinating but also shows extraordinary sophistication. When I found out that the author was only 17 years old, I was even more amazed.”

This discovery not only debunks a decades-old conjecture, but also opens up new approaches to many problems in harmonic analysis. “From now on, any similar problem, we will try to verify it with the Cairo-type structure,” added Oliveira.

The rejection of the Mizohata-Takeuchi conjecture also had a major consequence: the Stein conjecture, which was expected to bridge many other problems, was now forced to be reconsidered. Mathematics, once again, had to adjust its direction.

Skip bachelor's degree to go straight to doctorate

According to Quanta Magazine, after the results were announced, Cairo decided to skip her undergraduate degree and go straight to her doctorate. Of the 10 universities she applied to, most rejected her because Cairo did not have a high school or university degree. Two schools initially accepted her but were later blocked by administrators.

In the end, only the University of Maryland and Johns Hopkins accepted her. Cairo chose Maryland, where she will begin her doctoral journey this fall — her first formal degree.

(Source: Vietnamnet)

Link: https://vietnamnet.vn/thieu-nu-17-tuoi-giai-ma-duoc-bi-an-toan-hoc-ton-tai-suot-40-nam-2432764.html

Source: https://vtcnews.vn/thieu-nu-17-tuoi-giai-ma-duoc-bi-an-toan-hoc-ton-tai-suot-40-nam-ar960239.html