Seminar on OpenAI's 722 Manuscripts
Introduction to the Kakeya Conjectures
The Kakeya Conjectures concern overlapping properties of direction-separated tubes in $\mathbb{R}^n$. They are elementary to state, but difficult to prove. In 2025, Wang-Zahl proved the Kakeya Set Conjecture for $n=3$. On October 6, OpenAI claimed progress on Kakeya Conjectures in dimensions 3 and 4. Then on October 25, OpenAI claimed to have resolved these conjectures in all dimensions by proving Sogge's Local Smoothing Conjecture for free waves. I will introduce the Kakeya Conjectures and various human attempts, showing its many faces and why some approaches appeared difficult. I will also discuss how the Kakeya Conjectures connect to harmonic analysis and which directions mathematicians hoped a complete solution would unlock. Then I will share some personal first impressions on OpenAI's proposed proof for the Local Smoothing Conjecture on October 25.