Category: Miscellaneous
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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…
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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 Beautiful Paradoxes That Shape Mathematical Careers: When Less Truly Is More
It’s 3 AM and you’re staring at your to-do list: finish the differential geometry problem set, prep for tomorrow’s algebraic topology seminar, respond to that collaboration email, read three papers for your reading course, and somehow make progress on your own research. You tell yourself this is what serious mathematicians do, master everything, miss nothing,…
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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…
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The Secretary Problem and Academic Careers: Why Your Research Path is More Forgiving Than You Think
The Classic Problem: When You Only Get One Shot Imagine you’re a department chair tasked with hiring the best possible secretary from a pool of 100 applicants. The rules are harsh: you interview candidates in random order, must decide immediately after each interview whether to hire them, and once rejected, a candidate is gone forever.…