When Hannah Cairo started showing up at Berkeley Math Circle meetings as a teenager, nobody was betting she would topple a four-decade-old problem in harmonic analysis before she finished high school. At 17, she constructed a counterexample that challenges the Mizohata-Takeuchi conjecture, a long-standing prediction about how certain wave patterns can concentrate, and the work has had a ripple effect through the harmonic analysis community. Now 18 and on her way into graduate study, Cairo has become a local standout whose ideas are already reshaping how specialists think about restriction theory.
The paper and the preprint
Cairo detailed the result in a preprint titled "A Counterexample to the Mizohata–Takeuchi Conjecture," posted to the arXiv preprint server on Feb. 10. In it, she constructs a logarithmic ("log R")-loss counterexample for every C^2 hypersurface not contained in a hyperplane, using L^p estimates of an X-Ray transform to engineer fractal-style energy concentrations. According to arXiv, the construction demonstrates that Mizohata-Takeuchi-type bounds cannot be sharpened to improve endpoint multilinear restriction estimates.
Why the conjecture mattered
Posed in the 1980s, the Mizohata-Takeuchi conjecture had been viewed as a promising geometric strategy for attacking several endpoint problems in Fourier restriction theory. The intuitive story was that allowed waves should concentrate their energy along line-like sets in a fairly controlled way. Cairo's counterexample shows that this picture can break down in subtle, fractal fashions that slip past those expectations. As laid out by Quanta Magazine, her work is forcing researchers to rethink which technical approaches can survive in the presence of these new constructions.
Berkeley Math Circle roots
Cairo's route to the problem ran straight through local math culture. She first joined the Berkeley Math Circle as a student, later returning as an instructor, a combination that plugged her into campus researchers and graduate seminars. The Berkeley Math Circle lists her among its student-instructors, and founder Zvezdelina Stankova encouraged her to take graduate classes at UC Berkeley rather than immediately enrolling elsewhere. That concurrent-enrollment strategy is what ultimately landed Cairo in the course where the homework problem first appeared, according to The Daily Californian, and her listing as both student and instructor is confirmed on the people page for The Berkeley Math Circle.
A rapid ascent
Cairo was born in Nassau, the Bahamas, and was largely homeschooled. By the time she was 11, she had completed calculus, and by 14, she was applying to college, details she has discussed publicly. In a feature interview, she described how she moved from online self-study into Berkeley seminars and then turned a course homework problem into full-fledged research, as reported by Scientific American. This fall, the Davidson Institute named her a 2025 Davidson Fellow Laureate for the project, listing her hometown as Berkeley, along with her plan to begin graduate study at the University of Maryland, according to the Davidson Institute.
How the field reacted
Researchers who had pressed on Mizohata-Takeuchi for years met the news with a mixture of disbelief and excitement. "We were all shocked, absolutely," Itamar Oliveira told Quanta Magazine. Other analysts have described Cairo's construction as a surprisingly clean obstruction to the conjecture that nonetheless upends a popular strategy. Beyond the headlines, mathematicians expect the example to spark new work testing which weakened or "local" versions of the conjecture might still hold up against her counterexamples.
What's next
Cairo has said she plans to keep pushing on Fourier restriction problems, including a "local" Mizohata-Takeuchi formulation that could be compatible with her current constructions. In an interview with Scientific American, she described ongoing projects with her advisor, Ruixiang Zhang, and her personal website notes that she will begin graduate study at the University of Maryland this fall. The arXiv preprint and the seminars she has already given lay out concrete directions that other researchers will now test, refine, and extend as the field works through the implications of her result (arXiv).
