Episode 84 · The Machine's Theorems (Deep End, Ep 69–88), part 16 of 20
Shapes, Colours and Patterns: Convex Geometry and Combinatorics
Combinatorics is the part of mathematics where a twelve-year-old can understand the question and a professional can spend forty years on it. That makes it the perfect place to test what a machine-written proof is worth. Farah and Augustin colour the plane, squeeze convex bodies, hunt arithmetic progressions and tile space with a single strange block. Along the way we separate what OpenAI's manuscripts claim, what a Lean kernel has checked, and what the human community has actually said, including Joshua Swanson's verdict on the Shareshian–Wachs proof.
Listen to the show on your preferred podcast platform.
Inside this episode
- Cold open: the coloured tiles on the floor
- Where we are, and the rules of the road
- Convex bodies: what a mathematician means by a nice shape
- Mahler's volume product: a shape and its shadow-twin
- What family eighty-seven claims, and what Lean checked
- Brunn–Minkowski, and its logarithmic cousin
- Borsuk in dimension nine: when cutting a shape into smaller pieces fails
- Colouring the plane: the five-colour wall
- Arithmetic progressions: Erdosh's dream and Szemerédi's ladder
- Sidorenko, van der Waerden, and other counterexamples
- Tiling space with one strange block
- Kazhdan–Lusztig and Shareshian–Wachs: algebraic combinatorics
- Inside a Lean check, said in words
- Why so many, and why combinatorics?
- The human cost, in Tao's words and Swanson's
- Why it matters for an engineer
- Ledger and closing
Production credits & AI disclosure
Scripts generated with Claude Opus. Synthetic narration generated with Edge TTS. Reviewed, listened to, and approved by Sankalp Thakur.