Tuor Eladar

Hannah Cairo disproved a 40-year-old conjecture in harmonic analysis at age 17. In fall 2024 she emailed UC Berkeley professor Ruixiang Zhang and asked to sit in on his graduate course on Fourier restriction theory. She was not enrolled. Zhang said yes. The class centered on how waves and functions can be restricted to curved surfaces and what kinds of concentrations that produces. Zhang assigned a homework problem linked to the Mizohata-Takeuchi conjecture, an open question from the 1980s about weighted estimates for Fourier extension operators. For decades the field largely expected the conjecture to be true; a proof would have sharpened several other results. Cairo finished the assigned version, then kept working on the full statement. After months of unsuccessful attempts to prove it, she reversed direction and constructed a counterexample instead—using a projection of a higher-dimensional lattice (a hypercube construction) that produced a logarithmic loss where the conjecture predicted none. One counterexample is enough. She posted the paper, “A Counterexample to the Mizohata-Takeuchi Conjecture,” to arXiv on 10 February 2025. Zhang took weeks to be fully convinced; once he was, the result circulated quickly. Later that year Cairo presented it at a harmonic analysis conference in El Escorial, Spain. She and Zhang subsequently extended the work to power-loss counterexamples on dense families of convex hypersurfaces. Cairo was born in Nassau in 2007, homeschooled, and taught herself advanced mathematics largely alone after finishing calculus online by age 11. At 14 she joined the Berkeley Math Circle remotely; after the family moved to California she continued taking graduate courses through concurrent enrollment. She later applied directly to PhD programs, skipping the usual undergraduate and master’s degrees. Several departments turned her down for lack of a bachelor’s; the University of Maryland accepted her. She began doctoral work there in fall 2025 and received a $100,000 Davidson Fellows scholarship for the counterexample. The story is straightforward: a self-taught teenager treated a long-standing conjecture as a problem to be tested rather than an assumption to be proved, found a concrete case where it fails, and wrote it up. The mathematics community accepted the counterexample. She is now continuing work in Fourier restriction theory as a PhD student. 🌹