Hong Wang Wins 2026 Fields Medal, the Third Woman Ever
Shmerkin, from the University of British Columbia, who had additionally labored on the Falconer distance downside, observed Wang’s expertise whereas visiting Guth at MIT. She struck him as inventive, and as a sponge for brand new strategies. She appeared to grasp incidence geometry deeply although she had solely lately began engaged on it.
The analysis space was already replete with new instruments, and Wang quickly imported extra instruments from geometric measure concept that she had realized throughout a collaboration with Shmerkin on the Falconer distance set downside. She was vastly impressed by Shmerkin’s “multiscale analysis” strategy, a method of evaluating what occurs at totally different scales to be able to arrive at contradictions. “She is always very kind and acknowledges that some of the ideas came from my work, but she was able to apply them in these fantastic ways that I couldn’t have envisioned,” Shmerkin mentioned.
Wang’s status grew in 2019 when she, Guth, and Ruixiang Zhang of the University of California, Berkeley proved the 2D case of the local smoothing conjectureone in every of the holy grails of harmonic evaluation, which says that the resolution to the wave equation cannot focus power in a small area for an prolonged interval. The similar 12 months, her doctorate accomplished, she moved to Princeton, New Jersey, for a postdoc at the Institute for Advanced Study. She anxious that she could not succeed with out Guth, however progressively she started new collaborations and made headway on extra Kakeya-type issues. Hiring committees took discover; she received a college job at the University of California, Los Angeles in 2021, and in 2023 she turned an affiliate professor at NYU. A month after getting there, she produced another blockbuster result: Along with Kevin Renshe offered the 2D Furstenberg set conjecture, which estimates the dimensionality of a set of factors on the aircraft that accommodates a subset of traces pointing in each route.
Despite these successes, Wang nonetheless felt like an underdog. “Each time it just felt lucky,” she mentioned of her varied proofs.
It was at the Institute for Advanced Study throughout the pandemic that Wang had determined to show her consideration to the 3D Kakeya set conjecture. The downside had resisted many challengers over the a long time; proofs had even been introduced after which proven to be incorrect. “I never figured out the Kakeya conjecture, but there have been four or five times I thought I might have,” Guth mentioned. Each time, I’ve noticed his error earlier than going public.
Wang learn a 2014 weblog put up by the UCLA mathematician Terence Tao laying out a proof strategy that Tao and Katz had developed however by no means pursued. The strategy appeared promising, so she contacted Joshua Zahl, then at the University of British Columbia, one other mathematician who had studied points of the Kakeya downside in his thesis. They determined to research what Tao and Katz had described.
The technique, which wound up requiring an infinite arsenal of strategies, was to indicate that any hypothetical counterexample to the conjecture — a possible method of compacting line segments pointing in each route in 3D area into fewer than three dimensions — should have such a inflexible and environment friendly construction that it might contradict theorems governing the interaction between addition and multiplication. This common strategy of positing a counterexample that disproves some conjecture, then twisting it into such knots that you just conclude it will possibly’t exist in spite of everything (leaving the conjecture as the solely possibility), is a standard proof technique.
