| Throw out your mental image of the lone genius shouting "Eureka!" late at night. The real breakthroughs happen when mathematicians get together and exchange ideas.
Even when a landmark result is published by a single author, important moments of insight can usually be traced back to when they visited another department, heard a great talk at a conference, or had a seemingly irrelevant discussion. Many of math's great eureka moments happen IRL, wherever mathematicians gather.
Mathematicians felt this acutely when the Covid pandemic shuttered conferences for more than a year and prevented them from collaborating side by side at a chalkboard, requiring them to connect over Zoom instead. And these days, ChatGPT could pose a new threat — if, say, mathematicians decide to turn to it for a quick chat instead of knocking on a colleague's office door.
Whenever I start reporting a story, I anticipate the moment that I'll learn about the chance encounter that unstuck a thorny problem. There's almost always some run-in — at an airport, or a birthday party — without which the result would never have been possible. It's an important reminder that no discovery happens in a vacuum.
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The in-person math capital of the world is little more than a clearing in the dense Black Forest of southern Germany. I've heard about the legendary Oberwolfach Research Institute for Mathematics from so many mathematicians, and seen so many photos on their websites taken in front of its iconic statue, that it's made my own math bucket list. Quanta contributor Kevin Hartnett — my predecessor as math writer — was able to undertake this rite of passage in early 2020. He used the opportunity to chronicle a characteristic week of limited Wi-Fi and communal meals during a workshop on low-dimensional topology. His story is one of my favorites — a rare portrait of mathematics in practice that proves just how social it can be. A few months later, he followed up with a kind of requiem for collective mathematics in the Covid era, complete with photos of a locked-down Oberwolfach.
In-person math stories tend to share one common character: coffee. "A mathematician is a device for turning coffee into theorems," the mathematician Alfréd Rényi once said. I'm willing to wager that few pivotal moments in the history of modern math have happened without the drink's influence, whether or not it gets cited. For example, the mathematician Kathryn Mann didn't intend to start working on an important new proof when she visited Queen's University in Canada in 2019. But that's what ended up happening when she got coffee with Thomas Barthelmé. The caffeine boost, and the impromptu conversation it facilitated, ultimately led not to one paper but to many, as Quanta math editor Jordana Cepelewicz reported in 2023. The results put a long-overdue dent in the effort to mathematically describe an important class of dynamical systems called Anosov flows.
Another story stood out to me for its in-person moments, even if those moments didn't make it into the final draft. Nearly a year ago, I reported on research by Carlo Pagano and Peter Koymans, who had just published an important result about the limits of what mathematicians can ever know about certain equations. For a long time, the pair had been stuck. There seemed to be too many types of these equations to account for. Then came a game of mathematical telephone. In the early 2000s, the mathematician Sasha Shlapentokh had realized that if a method worked for a particular collection of equations, it would work for the rest. But she never published her finding. Then in 2017, she ran into another mathematician, Hector Pasten, at a colleague's birthday celebration and mentioned her result in passing. Seven years later, in 2024, it just so happened that Pagano and Pasten were in Toronto at the same time. They met up over a plate of dumplings, and Pasten passed along what he knew. Pagano had been about to give up on the problem. But with the dumpling-lunch knowledge in hand, he and Koymans were finally able to complete their proof.
"This is the sort of stuff that you do not get in the Zoom conferences," Pasten told me. "You really need to go for lunch and have dumplings with someone to realize what is the next trick for the next breakthrough."
Whatever tech revolutions the 21st century has in store, mathematicians will always need lunch, and they'll always be drinking coffee. So fortunately, the in-person magic won't be going away any time soon. | 
