# Microsoft Research — Solving a 31-year-old grand challenge in mathematics – the KLS conjecture

- Company: Microsoft Research (microsoft.com)
- Announced: 2026-10-21T16:00:00+00:00
- Category: research-paper
- Coverage: 1 outlet
- Announcement: yes
- Group: covered
- Source: https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/
- Record: https://forck.live/items/26034-solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture
- Subject: Research / Phi

Microsoft researchers and a collaborator have proved the Kannan-Lovász-Simonovits conjecture, a 31-year-old open problem in high-dimensional geometry, showing the bound holds with a fixed constant regardless of dimension. The KLS conjecture, proposed in 1995, asks whether the best flat or hyperplane cut of a broad class of high-dimensional distributions is nearly as effective as the best cut of any shape, differing by no more than a universal constant. Before this work, the best known result, established in July 2026, allowed that bound to grow slowly as dimensions increased. The authors improved that result using a human-directed agentic process in which AI agents explored more than 100 candidate approaches, many involving lengthy calculations with higher-order tensors, while the researchers evaluated, pruned and redirected the search. A new criterion was applied recursively, and in the October 1, 2026 version of the paper the dependence on dimension fell to the iterated logarithm, which still grows with dimension. Repeating the entire recursive refinement, an idea inspired by the authors' earlier work on fast linear programming algorithms, showed the accumulated errors stay bounded by a universal constant, completing the proof in the October 4, 2026 version. On the same day, a separate group of leading experts posted a full proof developed in parallel that built on the first version of the Microsoft paper, and a third proof relying on a closely related idea appeared separately.

## Evidence

Verbatim from https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/:

> The authors proved that the small errors introduced at each stage shrink fast enough that their total remains bounded by a universal constant, independent of the dimension. This eliminated the dependence on the number of dimensions and completed the proof in the October 4, 2026 version of the paper.

## Around this story

Outlets this record can name and link:

- Madrobot — Microsoft researchers say AI agents helped crack a famous 31-year-old maths problem: https://madrobot.blog/2026/10/21/microsoft-kls-conjecture-proof-ai-agents-31-year-old-maths-problem

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