{"id":"doi:10.5281/zenodo.21449565","kind":"article","title":"Proof of Geometric Langlands Conjecture Unifies Number Theory and Geometry — E8 Intelligence Research","authors":["Andrew Stewart Caldin"],"author_count":1,"year":2026,"published":"2026-07-20","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.21449565","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.21449565","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.21449565","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7169777280","pdf_url":null,"abstract":"FINDING: Proof of the geometric Langlands conjecture, a central pillar of the Langlands program, unifying number theory, geometry, and representation theory. | MATH: The geometric Langlands conjecture establishes an equivalence of derived categories: \\( D(Bun_G) \\cong QCoh(LocSys_{^LG}) \\), where \\( Bun_G \\) is the moduli stack of \\( G \\)-bundles on a Riemann surface and \\( LocSys_{^LG} \\) is the moduli stack of \\( ^LG \\)-local systems. This is a categorical Fourier–Mukai transform. Key constants: no specific numerical constants, but the dual group \\( ^LG \\) (Langlands dual) encodes root system symmetries (e.g., \\( G = SL_n \\) dual to \\( PGL_n \\)). | CONNECTION: Root systems and Weyl groups of Lie algebras (e.g., \\( E_8 \\), \\( E_7 \\), \\( E_6 \\)) are central. The dual group construction involves the Cartan matrix and Coxeter numbers (e.g., Coxeter number \\( h \\) for \\( E_8 \\) is 30). The moduli stacks involve affine Grassmannians and Hecke operators, which are deeply tied to affine root 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 \"Proof of Geometric Langlands Conjecture Unifies Number Theory and Geometry — E8 Intelligence Research\" by Andrew Stewart Caldin, https://doi.org/10.5281/zenodo.21449565, 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":[]}