Shmerkin, from the University of British Columbia, who had also worked on the Falconer distance problem, noticed Wang’s talent while visiting Guth at MIT. She struck him as creative, and as a sponge for new techniques. She seemed to understand incidence geometry deeply even though she had only recently started working on it.
The research area was already replete with new tools, and Wang soon imported more tools from geometric measure theory that she had learned during a collaboration with Shmerkin on the Falconer distance set problem. She was greatly inspired by Shmerkin’s “multiscale analysis” approach, a way of comparing what happens at different scales in order 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 said.
Wang’s reputation grew in 2019 when she, Guth, and Ruixiang Zhang of the University of California, Berkeley proved the 2D case of the local smoothing conjecture, one of the holy grails of harmonic analysis, which says that the solution to the wave equation can’t concentrate energy in a small region for an extended period. The same year, her doctorate complete, she moved to Princeton, New Jersey, for a postdoc at the Institute for Advanced Study. She worried that she couldn’t succeed without Guth, but gradually she began new collaborations and made headway on more Kakeya-type problems. Hiring committees took notice; she got a faculty job at the University of California, Los Angeles in 2021, and in 2023 she became an associate professor at NYU. A month after getting there, she produced another blockbuster result: Along with Kevin Ren, she proved the 2D Furstenberg set conjecture, which estimates the dimensionality of a set of points on the plane that contains a subset of lines pointing in every direction.
Despite these successes, Wang still felt like an underdog. “Each time it just felt lucky,” she said of her various proofs.
It was at the Institute for Advanced Study during the pandemic that Wang had decided to turn her attention to the 3D Kakeya set conjecture. The problem had resisted many challengers over the decades; proofs had even been announced and then shown to be wrong. “I never figured out the Kakeya conjecture, but there have been four or five times I thought I might have,” Guth said. Each time, he spotted his error before going public.
Wang read a 2014 blog post by the UCLA mathematician Terence Tao laying out a proof strategy that Tao and Katz had developed but never pursued. The approach seemed promising, so she contacted Joshua Zahl, then at the University of British Columbia, another mathematician who had studied aspects of the Kakeya problem in his thesis. They decided to investigate what Tao and Katz had described.
The strategy, which wound up requiring an enormous arsenal of techniques, was to show that any hypothetical counterexample to the conjecture — a potential way of squeezing line segments pointing in every direction in 3D space into fewer than three dimensions — must have such a rigid and efficient structure that it would contradict theorems governing the interplay between addition and multiplication. This general approach of positing a counterexample that disproves some conjecture, then twisting it into such knots that you conclude it can’t exist after all (leaving the conjecture as the only option), is a common proof strategy.