Solving a 31-year-old grand challenge in mathematics – the KLS conjecture
Microsoft Research Xinzhi Zhang
Microsoft says it proved a 31-year-old math problem about high-dimensional space. It could tighten the theory behind randomised algorithms and sampling.
Based on reporting by Microsoft Research, Xinzhi Zhang — read the original for the full story.
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Microsoft researcher Xinzhi Zhang and a collaborator say they have proved the Kannan-Lovász-Simonovits conjecture, a long-running problem from 1995 that sits at the center of high-dimensional geometry and probability. The claim matters because KLS is one of those results that quietly props up a lot of other work; if it holds, it gives a cleaner rule for how hard it is to move around in very high-dimensional spaces.
The conjecture asks whether, for a broad class of log-concave distributions, a simple flat cut is always almost as effective as the best cut of any shape, up to one universal constant. Microsoft’s team says the answer is yes. That means the geometry of these spaces is less wild than people had to assume before, and the hurdles facing randomized algorithms do not keep getting worse just because the dimension goes up.
The proof grew out of a human-directed process that used AI agents as explorers rather than as autopilot. The researchers say they tested more than 100 candidate approaches, with the AI systems generating possibilities, while the humans judged which paths to keep, drop, or push further. One of those paths produced a new criterion that became the key step in the final argument.
And the argument was not just iterative once; the researchers repeated the refinement itself, then showed the small errors at each stage stayed bounded by a universal constant. Earlier versions of the paper had only managed a bound that still crept upward, very slowly, with dimension. The final version, posted October 4, 2026, removes that dependence entirely.
Microsoft also points to outside validation. On the same day, other experts posted an independent proof developed in parallel, and a separate proof appeared as well. A Lean 4 formalization later followed, and the authors say they have shortened their own proof and checked that streamlined version too. This is the rare sort of math result that gets stronger as more people try to break it.
My take — AI-written commentary, not fact-checked reporting
The useful headline here isn’t “AI solved math.” It’s that AI is becoming a very expensive intern with infinite stamina, while the actual breakthrough still comes from human taste, judgment, and veto power. That’s the part the hype machine always skips, because “the model found it” sounds cleaner than “smart people used machines without surrendering the wheel.”
Read more about this at: Microsoft Research