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Research dossier · version 1 · updated Oct 30, 2026

Landmark Results in Pure Mathematics: State of the Field (October 2026)

This dossier surveys recent landmark results in pure mathematics: the geometric Langlands proof series, the three-dimensional Kakeya theorem, optimal sphere packings in dimensions 8 and 24, and the 2025 counterexample to additivity of the unknotting number. The main takeaway is that the newest headline results are preprints or unrefereed announcements, while the dimension 8 and 24 sphere-packing theorems are the most firmly settled.

3,405 words36 sources30 evidence-mapped claims7 benchmarks19 timeline events✓ citation-verified Oct 30, 2026

AI-assisted synthesis of the cited literature, independently fact-checked against its sources; not peer reviewed.

Contents

  1. Executive summary
  2. Background and core concepts
  3. Timeline of key milestones
  4. State of the art
  5. Geometric Langlands
  6. Kakeya conjecture
  7. Sphere packing
  8. Unknotting number
  9. AI-attributed proof reports
  10. Quantitative benchmarks
  11. Open problems and active debates
  12. Major programs and players
  13. Outlook: what to watch (near term)
  14. Key papers
  15. Glossary

Sample findings

  • Gaitsgory, Raskin and collaborators present a 2024 five-part preprint series as a complete proof of the geometric Langlands conjecture in characteristic-zero (de Rham) settings.
  • Subadditivity holds: the unknotting number of a connected sum is at most the sum of the unknotting numbers of the summands.
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