AI Reports · Microsoft · 2026-10-21 · Blog / commentary
Solving a 31-year-old grand challenge in mathematics – the KLS conjecture
A Microsoft researcher and a collaborator have resolved the Kannan–Lovász–Simonovits (KLS) conjecture, a 31-year-old problem proposed in 1995 that was widely regarded as one of the central open questions in high-dimensional geometry. The result establishes a universal geometric principle for log-concave distributions, showing that simple flat cuts capture their narrowest bottlenecks within a fixed factor regardless of dimension. The breakthrough emerged from a human-directed agentic research process in which AI systems explored possibilities while researchers evaluated and redirected the search, and it has been independently confirmed and followed by machine-checked formal verification.