Geometric Langlands Correspondence Proven via Derived Equivalence — E8 Intelligence Research

FINDING: Gaitsgory and collaborators have established the geometric Langlands correspondence as a derived equivalence between IndCoh and QCoh on the moduli stack of local systems, with recent 2024 lectures detailing the proof via singular support and 2-Fourier-Mukai transforms. | MATH: The core statement is an equivalence of derived categories: **IndCoh_N(𝒳) ≃ QCoh(𝒳^∨)** where 𝒳 = LocSys_G(Σ) (moduli of G-local systems on a curve Σ) and 𝒳^∨ = LocSys_{^L G}(Σ) (Langlands dual group ^L G). The proof uses **singular support** — an invariant assigning to each coherent sheaf a conical Lagrangian in the cotangent stack T*𝒳 — and a **2-Fourier-Mukai transform** acting on the derived category of sheaves with prescribed singular support. Key technical objects: the **categorical trace** Tr(Cat) and the **Drinfeld center** of the monoidal category of sheaves on the affine Grassmannian. No explicit numerical constants appear; the structure is categorical. | CONNECTION: The Langlands dual group ^L Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com