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. Chapters 00:00:00 Cold open: the coloured tiles on the floor 00:02:15 Where we are, and the rules of the road 00:04:47 Convex bodies: what a mathematician means by a nice shape 00:06:44 Mahler's volume product: a shape and its shadow-twin 00:10:32 What family eighty-seven claims, and what Lean checked 00:13:30 Brunn–Minkowski, and its logarithmic cousin 00:17:21 Borsuk in dimension nine: when cutting a shape into smaller pieces fails 00:21:07 Colouring the plane: the five-colour wall 00:24:39 Arithmetic progressions: Erdosh's dream and Szemerédi's ladder 00:28:51 Sidorenko, van der Waerden, and other counterexamples 00:32:25 Tiling space with one strange block 00:35:35 Kazhdan–Lusztig and Shareshian–Wachs: algebraic combinatorics 00:39:10 Inside a Lean check, said in words 00:42:14 Why so many, and why combinatorics? 00:44:15 The human cost, in Tao's words and Swanson's 00:46:41 Why it matters for an engineer 00:49:00 Ledger and closing Created by Sankalp Thakur. A Transient Labs production. Website: https://transientlabs.ai GitHub: https://github.com/sankalpsthakur X / Twitter: https://x.com/sankalpsthakur LinkedIn:   / sankalpthakur   Hugging Face: https://huggingface.co/sankalpsthakur Scripts generated with Claude Opus. Synthetic narration generated with Edge TTS. Reviewed, listened to, and approved by Sankalp Thakur.