What Is the Positive Grassmannian and Why Does It Show Up Everywhere?

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.   Note: Since this conversation was recorded, results from the First Proof, Second Batch project were released on June 10 2026.