Tag: Research Advice
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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…
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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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Navigating the Tension: Goal-Setting vs. Exploration in Mathematical Research
Recently, I’ve been thinking about a fundamental tension that every mathematician faces: the conflict between systematic goal-setting and open-ended exploration. This tension became particularly clear to me after reading Kenneth Stanley and Joel Lehman’s book “Why Greatness Cannot Be Planned: The Myth of the Objective.” The Stepping Stone Problem Stanley and Lehman argue that truly…
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The Mathematician’s Paradox: Why Chasing Trends Might Create More Lasting Mathematics Than Pursuing Eternal Truths
A counterintuitive guide to navigating academic mathematics using the Lindy Effect Every mathematics graduate student eventually faces the same existential question: Should I work on classical problems that have captivated mathematicians for centuries, or chase the latest trends, be they in machine learning, quantum computing, or cryptography? The answer, surprisingly, might challenge everything you’ve been…
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The Politics of Giving Credit
I recently had an unusual experience. I received critical feedback about a paper; not for failing to cite someone, but for citing them too generously. A colleague objected to a sentence in my introduction that described another researcher’s contributions as “important” to a body of theory. The objection wasn’t that the characterization was inaccurate. It…
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The Mathematical Proof That You Can’t Have It All: Arrow’s Impossibility Theorem and the Academic Career
Or: Why Your Career Anxieties Are Mathematically Inevitable Every mathematician knows that moment. You’re staring at your desk, covered with: Which do you tackle first? More fundamentally: How do you build a career that optimizes across all these dimensions? Here’s the liberating, terrifying truth: You can’t. And I mean that literally, as in, mathematically proven…