Category: Career Advice
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Sounding Sure Is Not the Same Skill as Being Right
I’ve sat on grant panels where I watched two proposals land on the table back to back. One argued, with total fluency, for a result that was, if I’m honest, more hope than mathematics. The other was written by someone who understood the risks in their own approach better than anyone else in the room,…
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Forty-One Hours: What Two Riesz Transform Papers Tell Us About AI and the Pace of Mathematics
On August 18, 2026, Yuyuan Ouyang, Daniel Spector, and Cody Stockdale posted a paper to arXiv settling a problem Elias Stein posed at the 1986 International Congress of Mathematicians: the best constant in the weak-type (1,1) bound for the vector Riesz transform on R^n is at most 2, independent of dimension. Forty years of open…
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The Four Laws: From Thermodynamics to Mathematical Research
There’s something deeply satisfying about the number four in physics. The four laws of thermodynamics represent some of the most fundamental principles governing our universe. But what’s truly remarkable is how these laws appear, sometimes literally, sometimes metaphorically, in completely different domains. Let’s explore three “four laws” frameworks: classical thermodynamics, black hole thermodynamics, and a…
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The Cost of Sitting on a Result
A while back I was finishing a paper with a few coauthors, extending an earlier result of ours to a more general setting. The key step, the thing that made the whole proof work, was, once you saw it, almost embarrassingly simple to state. It reframed a technical problem as something much more standard, and…
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What Does the AI Proof of Crouzeix’s Conjecture Mean for Research Mathematics?
The recent solution of Crouzeix’s conjecture is remarkable mathematics. But the story of how the proof was found may be even more consequential for mathematicians. As described by Alex Townsend and Anne Greenbaum in SIAM News, Shanmu Jin, a neurosurgery resident and postdoctoral researcher with relatively little formal mathematical training, used GPT-5.6 Sol in ChatGPT…
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The Habit Nobody Wants
What a self-taught miller can tell us about what actually separates successful researchers from everyone else. Nottingham, sometime around 1823. A miller named George Green climbs the stairs of his father’s windmill after a full day’s work, hauling sacks, tending the grindstones, breathing flour dust, and sits down under a candle with a borrowed mathematics…
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Embrace the Suck
“Embrace the suck” is military slang for accepting that some situations are going to be difficult, uncomfortable, and uncertain, and learning to operate effectively anyway. It’s about finding ways to function when conditions are far from ideal, when you don’t have complete information, and when there’s no clear path forward. Mathematical research is full of…
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The Mathematical Sherlock Holmes
“I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose… It is a mistake to think that that little room has elastic walls and can distend to any extent. Depend upon it there comes a time when for every addition…
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The Mathematical Polyglot
For the second post in the series look here. The most profound mathematical insights often come from connecting ideas across different fields. But here’s the challenge: mathematical communities speak different languages. An algebraist and an analyst might as well be from different countries, despite often studying closely related structures. Think of mathematical fields as linguistic…
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What a Chess Engine Knows About Taste
A graduate student wrote to me recently, after reading something I’d posted about how the value in our work is migrating from execution toward judgment. He’d been circling the same thought: that the future of pure mathematics may be governed less by the practical pressures that shaped the profession for the last century and more…