What Is the Positive Grassmannian and Why Does It Show Up Everywhere? | Quanta Magazine
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What Is the Positive Grassmannian and Why Does It Show Up Everywhere?
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The Joy of Why
What Is the Positive Grassmannian and Why Does It Show Up Everywhere?
By
Janna Levin
and
Steven Strogatz
June 25, 2026
Lauren Williams tells 'The Joy of Why' how studying a fundamental object in algebraic combinatorics led to a career full of surprises.
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Chanelle Nibbelink for Quanta Magazine
Introduction
Authors
Janna Levin
Contributing Columnist
Steven Strogatz
Podcast Host
June 25, 2026
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algebraic geometry
amplituhedron
combinatorics
mathematical physics
mathematics
The Joy of Why
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What links certain mathematical models of traffic flow, shallow-water waves, and quantum particle scattering? The surprising answer lies in a corner of the algebraic combinatorics world that goes by the name of positive Grassmannian. In simple terms, the positive Grassmannian is a shape that classifies other shapes. Remarkably, pieces of the positive Grassmannian can be reassembled in forms that reveal shared structures in these and many other seemingly unrelated mathematical systems.
That we know the positive Grassmannian crops up in many real-world settings is largely down to the theoretical work of Lauren Williams at Harvard University. In this latest episode of The Joy of Why, Williams talks to co-host Steven Strogatz about her work, how she realized the surprising pervasiveness of the positive Grassmannian, and how she has made a career of finding connections among fields that don’t at first sight seem connected. The conversation then switches to another project Williams is working on, called First Proof, which is trying to measure objectively how good AI systems are at coming up with proofs of research-level mathematical statements, and which leads to an exploration of whether AI may or may not take over mathematics.
Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.
Note: Since this conversation was recorded, results from the First Proof Second Batch project were released on June 10, 2026.
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[Music plays]
STEVE STROGATZ: All right.
JANNA LEVIN: Okay, now we’re starting, starting?
STROGATZ: I’m Steve Strogatz,
LEVIN: And I’m Janna Levin.
STROGATZ: And this is The Joy of Why.
LEVIN: A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today.
STROGATZ: I can start us off with a discussion I had recently with a mathematician who works in the area of math that we call algebraic combinatorics.
LEVIN: Oof.
STROGATZ: Yeah? What makes you say “oof” to that?
LEVIN: Oh, I don’t know. It sounds really hard. I mean, algebraic, we know what those two words mean. Combinatorics, I assume some kind of discrete algebra.
STROGATZ: Yeah, something like that, right? It’s all about discrete objects or structures.
LEVIN: When I think of combinatorics, I sometimes think of the many ways in which you can organize something. Like this comes up in thermodynamics where statistical thinking or in counting of entropy. What are the possible ways of organizing a sequence of numbers, for instance? And is that an element in combinatorics, or is that too simplistic?
STROGATZ: No, that’s exactly on the money. That’s the kind of thing that we’re thinking about. And for our listeners who may not be in the world of thermodynamics or statistical physics, we do combinatorics any time we play card games.
LEVIN: Right, exactly.
STROGATZ: You know, so you can ask yourself, with a deck of 52 cards and the four suits, how many ways are there to get three of a kind versus two of a kind or a straight? Those kinds of counting questions or probability questions, that’s part of what gave rise to combinatorics, but now it’s used all the time when we have to find clever time-saving ways to count things or to organize things, as you said, in some very structured manner.
LEVIN: Yeah, that’s a great analogy. That’s exactly how I think of it. If you’re handed five cards, what are your possible combinatorics? Your first card is a king, then what are the possible combinations that follow with the next four cards, and then the next three, and then the next two, and then the...