The Geometric Langlands Program

The Geometric Langlands Program represents one of the most ambitious unification projects in modern mathematics, connecting algebraic geometry, number theory, and representation theory through the revolutionary lens of sheaf theory and moduli spaces.

Core Mathematical Framework

Fundamental Correspondence

The geometric Langlands correspondence establishes a duality between:

  • D-modules on the moduli space of G-bundles on a curve X
  • Quasi-coherent sheaves on the moduli stack of ^LG-local systems on X

Geometric Langlands Conjecture

For a reductive group $G$ and its Langlands dual $^LG$, there exists an equivalence of categories:

$$\text{D-mod}(\text{Bun}_G) \cong \text{QCoh}(\text{LocSys}_{^LG})$$

Interactive Concept Network

Moduli Space of Vector Bundles

Sheaf Theory Visualization

Presheaf Structure

Sheafification Process

Leading Research Papers

Geometric Langlands Correspondence for D-modules in Prime Characteristic
D. Gaitsgory, D. Arinkin (2024)
arXiv:xxxx.xxxxx

Revolutionary work extending the correspondence to positive characteristic settings.

Quantum Geometric Langlands Correspondence
E. Frenkel, D. Gaitsgory (2023)
arXiv:xxxx.xxxxx

Explores quantum deformation of the geometric Langlands program.

Categorification of the Geometric Langlands Program
D. Ben-Zvi, J. Francis, D. Nadler (2023)
arXiv:xxxx.xxxxx

Higher categorical approach to geometric Langlands using derived algebraic geometry.

Spectral Decomposition and the Geometric Langlands Conjecture
V. Lafforgue (2023)
arXiv:xxxx.xxxxx

Breakthrough on spectral decomposition in the function field case.

Leading Research Centers

  • Institute for Advanced Study (IAS) - Home to Edward Witten, Dennis Gaitsgory
  • Harvard University - Dennis Gaitsgory's group
  • University of Chicago - Geometric representation theory group
  • Paris-Saclay - French school of algebraic geometry
  • Perimeter Institute - Mathematical physics connections

Current Research Frontiers

Categorical Geometric Langlands

Extension to derived categories and higher categorical structures, incorporating tools from ∞-category theory.

Quantum Geometric Langlands

Deformation of the correspondence involving quantum groups and their representations.

Arithmetic Applications

Connections to the classical Langlands program and applications to number theory.