Needles, Waves and Dust: Harmonic Analysis Falls

Deep End Era. Engineering with Sankalp · Oct 12, 2026 · 53m

Part 15 of "The Machine's Theorems". Farah and Augustin start with Kakeya's needle, turning a unit segment in every direction using as little room as possible, then climb to Fourier restriction, Bochner–Riesz summability, Sogge's local smoothing conjecture for the wave equation, Falconer's distance sets and Carleson's pointwise convergence problem. We read OpenAI's manuscripts in families 074 (the Kakeya maximal conjecture in 3D, building on Wang and Zahl's 2025 set theorem, and Kakeya sets in 4D), 077 (restriction), 078 (Bochner–Riesz in 3D), 079 (local smoothing in 3D), 073 (Falconer in all…