Yesterday was one of the biggest days in mathematical history. Among 372 major results released by OpenAI, on the recommendation of an advisory group that included Timothy Gowers and Edward Witten, was a proof of Subhash Khot's Unique Games Conjecture (UGC). The conjecture implies that a whole slew of optimization problems are NP-hard, even for approximations slightly better than those given by semidefinite programming relaxation. Complexity theorist Dana Moshkovitz, who has worked toward proving the UGC for her entire career, found the news both vindicating and humbling.
There is a Lean certificate for the proof, as there is for some of the other results, but few humans have yet understood the proofs. The race to do so has just begun. Moshkovitz described the UGC paper as horribly written and said it was nearly impossible to read without AI help. She noted that the proof invents a bizarre new code with a noise test, and that many citations seem irrelevant or confusing.
Other treasures from the release include L=BPL, showing that probabilistic and deterministic logspace are the same, one of the great derandomization conjectures short of P=BPP. Another result improves the running time of the Fourier Transform and integer multiplication to O(n log^0.9999999999999 n), breaking a barrier that had stood since the 1960s. A positive solution to the Unitary Synthesis Problem, posed in 2007, also appeared.
For mathematicians and theoretical computer scientists, the moment is both exciting and unsettling. A possible future is a math world that is heavenly for those with creative ideas that AI can help check and implement. As the author notes, there is a lot for humans to learn from these new results.
Source: Hacker News · Summarized by HeadlinesBriefing