Shmerkin, from the College 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 spanking new methods. She appeared to grasp incidence geometry deeply regardless that she had solely not too long ago began engaged on it.
The analysis space was already replete with new instruments, and Wang quickly imported extra instruments from geometric measure principle that she had realized throughout a collaboration with Shmerkin on the Falconer distance set downside. She was significantly impressed by Shmerkin’s “multiscale evaluation” strategy, a method of evaluating what occurs at completely different scales with the intention to arrive at contradictions. “She is all the time very form and acknowledges that a number of the concepts got here from my work, however she was in a position to apply them in these unbelievable ways in which I couldn’t have envisioned,” Shmerkin stated.
Wang’s popularity grew in 2019 when she, Guth, and Ruixiang Zhang of the College of California, Berkeley proved the 2D case of the native smoothing conjecture, one of many holy grails of harmonic evaluation, which says that the answer to the wave equation can’t focus power in a small area for an prolonged interval. The identical yr, her doctorate full, she moved to Princeton, New Jersey, for a postdoc on the Institute for Superior Research. She anxious that she couldn’t succeed with out Guth, however regularly she started new collaborations and made headway on extra Kakeya-type issues. Hiring committees took discover; she bought a school job on the College of California, Los Angeles in 2021, and in 2023 she grew to become an affiliate professor at NYU. A month after getting there, she produced one other blockbuster outcome: Together with Kevin Ren, she proved the 2D Furstenberg set conjecture, which estimates the dimensionality of a set of factors on the airplane that incorporates a subset of traces pointing in each course.
She’s loopy into math. She’s positively one of the vital devoted individuals I’ve seen.
Shukun Wu
Regardless of these successes, Wang nonetheless felt like an underdog. “Every time it simply felt fortunate,” she stated of her numerous proofs.
It was on the Institute for Superior Research in the course of the pandemic that Wang had determined to show her consideration to the 3D Kakeya set conjecture. The issue had resisted many challengers over the many years; proofs had even been introduced after which proven to be incorrect. “I by no means discovered the Kakeya conjecture, however there have been 4 or 5 occasions I assumed I may need,” Guth stated. Every time, he noticed his error earlier than going public.
Wang learn a 2014 weblog submit by the UCLA mathematician Terence Tao laying out a proof technique that Tao and Katz had developed however by no means pursued. The strategy appeared promising, so she contacted Joshua Zahl, then on the College 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 methods, was to point out that any hypothetical counterexample to the conjecture — a possible method of compacting line segments pointing in each course in 3D area into fewer than three dimensions — should have such a inflexible and environment friendly construction that it could contradict theorems governing the interaction between addition and multiplication. This basic strategy of positing a counterexample that disproves some conjecture, then twisting it into such knots that you just conclude it could possibly’t exist in spite of everything (leaving the conjecture as the one possibility), is a standard proof technique.
