{"id":"doi:10.5281/zenodo.23310227","kind":"article","title":"Geometric Langlands Correspondence Proved, Unifying Number Theory, Analysis, and Geometry — E8 Intelligence Research","authors":["Andrew Stewart Caldin"],"author_count":1,"year":2026,"published":"2026-10-12","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.23310227","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.23310227","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.23310227","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7221544160","pdf_url":null,"abstract":"FINDING: The Langlands program — a conjectural web linking number theory, harmonic analysis, and geometry — has seen a major recent proof in its geometric form, unifying disparate mathematical realms via automorphic forms and Galois representations. | MATH: Core objects: automorphic forms (e.g., modular forms, Maass forms), L-functions (Euler products, functional equations), Galois representations (ρ: Gal(ℚ̄/ℚ) → GLₙ(ℂ)), and the Langlands correspondence: π ↔ ρ, where π is an automorphic representation and ρ a Galois representation. Key recent result (Gaitsgory et al.): proof of the *geometric* Langlands conjecture — an equivalence of derived categories: D-modules on Bun_G ↔ IndCoh(LocSys_G^∨), where G is a reductive group and G^∨ its Langlands dual (root system inversion: roots ↔ coroots). | CONNECTION: The Langlands dual group G^∨ is defined by *inverting the root system* — this is a crystallographic symmetry (Weyl group action, Cartan matrix transposition). For G = GLₙ, G^∨ = GLₙ (s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com","abstract_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex","attribution":"Abstract of \"Geometric Langlands Correspondence Proved, Unifying Number Theory, Analysis, and Geometry — E8 Intelligence Research\" by Andrew Stewart Caldin, https://doi.org/10.5281/zenodo.23310227, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}