| Often, all it takes to break new mathematical ground is a fresh set of eyes. When a problem stands unvanquished for decades or centuries, it's likely that experts in the area have already tried to tackle it with every technique they know. Established approaches won't do the job.
Sometimes, in cases like these, the problem ultimately gets solved not by the experts who have spent years chipping away at it, but by a total newcomer — someone who can bring new perspective and insight, either from a totally different area of mathematics or from outside academia altogether. Someone with no prior connection to a topic will occasionally arrive at the right moment, carrying just the right tool kit, to fell a major mathematical question with a single blow.
In math, where subfields are highly specialized and can often be navigated only after years of training, progress tends to occur gradually — in a trickle of intermediate proofs written by a small community of insiders. So these sudden injections of novelty are particularly exciting, heralding an influx of ideas that can energize an area of study and lead to all sorts of new mathematics.
These moments also illustrate an important lesson: that a proof can come from anywhere. If its steps are all correct, then it doesn't matter whose name is at the top and what their affiliation is. Being well known may help build trust in a result, but it's ultimately the truth that mathematicians care about.
Let's take a look at some of the times when an amateur or outsider made a major mathematical advance. These stories are a chance to celebrate the connective power of math, and to inspire those who hope to contribute to its grand project, whatever their background.
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Hannah Cairo followed an unconventional path into the math world. As Kevin Hartnett recounted in a wonderful story this summer, she was homeschooled in the Bahamas, and by 14 she had reached the level of an advanced undergraduate math major — by learning math on the online learning platform Khan Academy. In February, she solved the Mizohata-Takeuchi conjecture, which is about functions built out of waves that lie on a sphere or some other geometric surface. A property called the "energy" of these functions, according to the conjecture, is supposed to spread out and concentrate in certain patterns. But Cairo found a novel example where the energy instead coalesces to form a strange, fractal-like pattern — a counterexample that more seasoned mathematicians had missed for 40 years. This fall, Cairo started her graduate studies at the University of Maryland — one of two programs to accept her, even though she has neither a high school diploma nor a college degree. She's 18.
A more common "outsider" contribution takes place when an established mathematician wanders into an unfamiliar field. I wrote earlier this year about Boaz Klartag, who until a year ago knew essentially nothing about sphere packing, a problem about how many spheres you can pack into a box. His studies instead focused on the geometry and symmetries of high-dimensional convex shapes. But he knew that a century ago, a couple of mathematicians had related sphere packing in higher dimensions to these other shapes. So he started reading up on sphere packing, and within a few months he'd demolished a barrier to progress on the problem that had stood for decades. Some people think that the arrangement of spheres that Klartag found is the best possible packing for arbitrarily high dimensions, full stop. But that remains to be proved.
Sometimes the unfamiliar wanderer comes from outside math altogether. On its face, the Connes embedding conjecture has nothing to do with computer science or physics. It sounds like pure math, because it is. It says, among other things, that you can often use large finite tables of numbers called matrices to approximate infinite ones. But it was a team of computer scientists that solved it in the end. They had set out to investigate when it's possible to verify solutions to computational problems using a quantum computer, and when it's not. In doing so, Quanta reported in 2020, they disproved a conjecture in physics about how to mathematically model entanglement — and also disproved the Connes embedding conjecture. It's an example of how surprising results in math often come from connections to other fields of research.
And then there's the world of tiling. Out of all the math stories Quanta covers, the ones that most often feature amateur mathematicians are the ones about tiling problems. Anyone who's stared idly at a tiled bathroom floor or played too much Tetris has pondered which kinds of shapes can fit together to fill a space without gaps or overlaps. So it may not come as a surprise that a five-decade hunt for the particularly elusive "einstein" tile was ended by a jigsaw puzzle–obsessed retiree named David Smith. Longtime contributor Erica Klarreich covered his achievement for Quanta in 2023. His tile resembled a hat and was an instant sensation. Mathematicians who had sought such a shape their whole career found its simple contours "just mind-boggling." | 
