Microsoft Resolves 31-Year-Old KLS Conjecture

Microsoft researchers have solved a long-standing mathematical problem in high-dimensional geometry and probability.

By Precis Daily NewsroomOctober 23, 20261 min read181 words

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What happened

Microsoft researchers, including Xinzhi Zhang, have addressed a longstanding problem in mathematics known as the KLS conjecture. This problem, which has remained unsolved for 31 years, concerns high-dimensional geometry and probability theory.

The resolution of the KLS conjecture builds upon a prior result from July 2026. Microsoft's team further refined this work to enhance the mathematical bound, significantly reducing the dependence on dimension to the iterated logarithm.

Why it matters

The KLS conjecture has implications for various fields, including theoretical computer science and statistics. Solving it provides a deeper understanding of how random processes behave in high-dimensional spaces, which is relevant to algorithm design and data analysis.

Key details

The final step in the proof involved repeating a recursive refinement process, which was inspired by earlier work on fast algorithms for linear programming. This approach allowed the researchers to achieve a more precise and efficient solution.

The authors have simplified their proof and verified the streamlined argument using the Lean proof assistant. They plan to release both the proof and the verification after completing final reviews.