Tag: Research Advice
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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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You Are Who You Work With
A folk saying, a body of research, and what a biology lab can teach mathematicians about doing their best work. Jim Rohn, the American entrepreneur and motivational speaker, had a saying he returned to often: “You are the average of the five people you spend the most time with.” It is the kind of aphorism…
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The Infinite Monkeys Aren’t Random Anymore
When OpenAI announced that one of its systems had disproved a long-standing conjecture in combinatorial geometry, the headline wrote itself: an AI solved a famous open problem in mathematics. The story traveled fast, and the framing was irresistible, a machine had done the thing we’d been told machines couldn’t do. I read the paper, along…
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The Microscope and the Telescope: A Defensive Lineman’s Advice for Mathematicians
Maxx Crosby, the Raiders’ star pass rusher, described his approach to his career in a 2024 interview with Sports Illustrated using a metaphor that’s stuck with me. He credits it to two of his former coaches, Rich Bisaccia and Rod Marinelli: The microscope’s your day-to-day; your 99% of the time, you’re in today. All you’re…
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The Complete Guide to Losing the Academic Mathematics Metagame
A comprehensive manual for sabotaging your own mathematical career Every game has a metagame, the game about the game, the optimal strategies for winning at a specific point in time. Academic mathematics is no exception. While most career advice tells you how to succeed, I’m going to do something different: show you exactly how to…
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The White Whale: On Finding the Problem That Hunts You Back
A letter to graduate students, postdocs, and mathematicians at every stage, on obsession, identity, and the problem you cannot let go. “To the last I grapple with thee; from hell’s heart I stab at thee; for hate’s sake I spit my last breath at thee.” — Herman Melville, Moby-Dick (1851) Somewhere in your mathematical life,…
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When Everything Becomes a Crab: The Strange Phenomenon of Convergent Evolution in Nature, Mathematics, and Human Systems
The Crab at the End of the Universe There’s a running joke among evolutionary biologists that given enough time, everything wants to become a crab. This isn’t entirely hyperbole, at least five separate groups of crustaceans have independently evolved into crab-like forms over millions of years. This phenomenon, called carcinization, has become something of an…
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Give Me a Lever Long Enough
“Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” —Archimedes Archimedes was talking about physics, but academics have always understood the metaphor intuitively. We speak of “high-leverage activities,” of people who “punch above their weight,” of the importance of “strategic positioning.” We recognize that…
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The Energy to Begin
On activation energy, steady state, and the hidden structure of a research career In chemistry, activation energy is the minimum energy required to initiate a reaction. Once that threshold is cleared, the reaction often sustains itself at a considerably lower cost, what physicists might call a steady state. The barrier to starting is higher, sometimes…