Lauren Williams tells The Joy of Why how studying a fundamental object in algebraic combinatorics led to a career full of surprises.
“The Joy of Why” is a Quanta Magazine podcast about curiosity and the pursuit of knowledge. The mathematician and author Steven Strogatz and the cosmologist and author Janna Levin take turns interviewing leading researchers about the great scientific and mathematical questions of our time. For more episodes visit Quanta Magazine: https://www.quantamagazine.org/tag/the-joy-of-why/
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CHAPTERS
00:00 Intro: algebraic combinatorics explained
04:13 Lauren Williams on mathematical beauty
08:03 Why combinatorics is becoming central to mathematics
14:45 Understanding the Grassmannian
22:08 Why the positive Grassmannian matters
24:00 Positive Grassmannian explained in detail
29:28 Why the Positive Grassmannian Appears Everywhere
29:28 Surprising connections to physics, traffic, and nature
38:01 The amplituhedron and particle physics
43:00 Can AI discover new mathematics?
49:57 Outro
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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.
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"The Joy of Why" is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine. The Joy of Why is produced by PRX Productions. The production team is Caitlin Faulds, Jade Abdul-Malik, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzales. Edwin Ochoa is our project manager.
From Quanta Magazine, Simon Frantz and Samir Patel provided editorial guidance, with support from Samuel Velasco, Simone Barr, and Michael Kanyongolo. Samir Patel is Quanta’s editor-in-chief. The episode art is by Chanelle Nibbelink, and our logo is by Jackie King and Kristina Armitage. Special thanks to Garth Avery at the Cornell Broadcast Studio.
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