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Homeschooled Student Solves 40-Year-Old Math Theory, Gets Direct PhD Admission
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Hannah has been offered a place in the University of Maryland's PhD programme in mathematics, bypassing the need for a high school diploma or college degree.
Anyone can earn a PhD if they complete their degree and postgraduate studies, pass difficult entrance exams, and qualify at the top. However, a young woman who studied at home rather than attending school got the chance to pursue a PhD directly after solving a mathematical theory that had been a mystery for 40 years. Seventeen-year-old Hannah Cairo from the Bahamas achieved this remarkable feat.
Hannah disproved the Mizohata–Takeuchi conjecture, a longstanding problem in mathematics within the realm of harmonic analysis, which studies how waves behave on curved surfaces. Many mathematicians had attempted to solve it over the years, but Hannah proved it wrong and provided a clear example to demonstrate why.
Born in the Bahamas and homeschooled by her parents, who encouraged her to delve deeply into subjects of interest, Hannah chose mathematics. By age 11, she had mastered calculus and went on to study university-level subjects such as linear algebra, differential equations, and topology. She read books, took online classes, and occasionally had tutoring.
During the COVID-19 lockdowns, Hannah joined the Chicago Math Circle online, where she engaged with problems more advanced than typical schoolwork. This introduced her to the questions and ideas that professional mathematicians grapple with.
At 14, Hannah applied to the Berkeley Math Circle summer programme, stating on her application that she had already covered the advanced undergraduate math curriculum—a true claim. Her application was accepted, allowing her to join Berkeley's concurrent enrolment program and take university-level classes alongside graduate students, all while still officially in school.
A professor named Ruixiang Zhang introduced Hannah to the Mizohata-Takeuchi conjecture, which predicted limits on how waves can focus on curved surfaces. While many students might have viewed it as a challenging puzzle, Hannah sought to create examples that defied the conjecture's rules, despite frequently encountering dead ends. Although Zhang doubted some of her ideas, Hannah persisted.
Her breakthrough came with a seemingly simple idea: instead of trying to work within the pattern of the conjecture, she thought of creating an entirely different wave pattern. She developed waves that extended rather than intersected. After thorough testing, her example showed that the conjecture's predictions were completely incorrect.
In February, Hannah published her research paper on arXiv. Some experts were astonished and congratulated her, while others were sceptical, questioning how someone so young could achieve such a feat. Consequently, Hannah was offered a place in the University of Maryland's PhD programme in mathematics, bypassing the need for a high school diploma or college degree.
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First Published:
August 13, 2025, 14:36 IST
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