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

Research dossier · v1 · updated 2026-10-30 · 3,405 words · 36 sources · 30 claims · verification: verified

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.

## Contents

- Executive summary
- Background and core concepts
- Timeline of key milestones
- State of the art
  - Geometric Langlands
  - Kakeya conjecture
  - Sphere packing
  - Unknotting number
  - AI-attributed proof reports
  - Quantitative benchmarks
- Open problems and active debates
- Major programs and players
- Outlook: what to watch (near term)
- Key papers
- 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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