Recent OpenAI Math dump have solutions to many long standing probems in Math and TCS. In the dump, two problems that I and my friends, notably Arnold Filtser, have studied for more than a decade, and published a few papers about this:
- The $\ell_1$-embedding conjecture for planar graphs. Open AI solution here.
- The $\ell_1$-embedding conjecture for bounded treewidth graphs. Open AI solution here.
It is unsettling and hard to swallow. I have not looked at the details yet, and will be doing so in the next few days. On a positive note, I hope to learn new techniques in planar graphs. I always believe that we have not been able to solve these problems because we lack a serious understanding of planar metrics. Now that they are solved, learning what the serious understanding is exciting.
A clear next prediction (not included among Open AI solution) is a solution of the conjecture that that minor-free graph metrics are embeddable into $\ell_1$ with constant distortion. Using the Robertson-Seymour decomposition, one basically could reduce this conjecture to bounded treewidth and planar graphs. At this point, I feel that understanding the two results above are more important than churning out another result.
Will udpate my understsanding of the two papers above.
I intentially do not mention other big results. What a strange time to be alive.