Category: Miscellaneous
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Borrowed Confidence, Earned Belief
Fake it, do it, make it: a mathematician’s path through impostor syndrome. I cannot tell you when it happened. There was no single moment, no paper accepted, no problem solved, no email from a senior colleague that flipped a switch. What I noticed, eventually, was that conversations had changed. My advisor would pause longer when…
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The Missing Dimensions
The first time you really understood a proof, you probably weren’t reading it. You were in the middle of something else, your own problem, your own mess of ideas, when the technique you’d read about three months earlier suddenly clicked into place. Not because you’d finally memorized it, but because you needed it. The stopping…
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Minor Revisions
A single peer review decision, seen from four angles, and what it says about a process that is slow, imperfect, and usually right. The email arrives on a Tuesday afternoon. You’ve spent three months revising. You addressed every point, the notation, the exposition, the logical gap flagged in section three. You submitted again. You were…
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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 Summit That Isn’t
On false peaks, hidden terrain, and how to navigate a mathematical career without being fooled by the view There is a particular cruelty built into mountain terrain. You climb for hours, legs burning, eyes fixed on the ridge above, and when you finally crest it, gasping, you discover not the summit but another slope rising…
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Zero to One in Mathematical Research: Beyond Incremental Progress
Peter Thiel’s Zero to One distinguishes between two types of progress: going from “1 to n” (horizontal progress through copying or incremental improvement) versus going from “0 to 1” (vertical progress through creating something entirely new). While Thiel wrote about startups and business, this framework offers a fascinating lens for understanding mathematical research. The Mathematics…
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A Mathematical Blockchain for Research Credit: Why It Won’t Happen (And What We Can Do Instead)
Mathematical research has a dirty secret: we’re terrible at sharing knowledge of failed approaches, and we’re not always honest about crediting the work that enables our breakthroughs. These problems are connected, and they’re costing us enormous intellectual progress. The Failure Gap When mathematicians publish papers, they share what worked. What they don’t share, what they…
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The Compound Advantage: Why Small Edges Become Insurmountable Leads
In 1968, sociologist Robert Merton noticed something peculiar: scientists who were already famous received disproportionate credit for discoveries, even when lesser-known researchers did similar work. He called this the “Matthew Effect,” after the biblical verse: “For to everyone who has, more will be given.” This wasn’t just about science. Merton had identified a fundamental pattern…