{"id":"doi:10.5281/zenodo.22122081","kind":"article","title":"Geometric Langlands Conjecture Proven: Unifying Arithmetic and Geometric Number Theory — E8 Intelligence Research","authors":["Andrew Stewart Caldin"],"author_count":1,"year":2026,"published":"2026-08-27","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.22122081","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.22122081","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.22122081","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7204471046","pdf_url":null,"abstract":"FINDING: The 2024 proof of the geometric Langlands conjecture establishes a categorical equivalence between D-modules on the moduli stack of G-bundles and ind-coherent sheaves on the moduli stack of local systems, unifying arithmetic and geometric number theory. | MATH: The core statement is an equivalence of derived categories: \\( \\mathrm{D-mod}(\\mathrm{Bun}_G) \\simeq \\mathrm{IndCoh}(\\mathrm{LocSys}_{G^\\vee}) \\), where \\(G\\) is a reductive group and \\(G^\\vee\\) is its Langlands dual (e.g., \\(SL_n \\leftrightarrow PGL_n\\)). This involves the categorical trace, Hecke eigensheaves, and the geometric Satake equivalence — a tensor functor \\( \\mathrm{Rep}(G^\\vee) \\to \\mathrm{Perv}(\\mathrm{Gr}_G) \\) on the affine Grassmannian. Key constants: the dual group's root system is the *dual* of \\(G\\)'s root system, swapping long and short roots — a symmetry operation that preserves the Weyl group order but flips Dynkin diagram arrows (e.g., \\(B_n \\leftrightarrow C_n\\)). | CONNECTION: The Langlands dua 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 Conjecture Proven: Unifying Arithmetic and Geometric Number Theory — E8 Intelligence Research\" by Andrew Stewart Caldin, https://doi.org/10.5281/zenodo.22122081, 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":[]}