A cross-cutting core objective synthesizing the local, global, and geometric branches of Langlands.

A cross-disciplinary synthesis problem connecting arithmetic and geometric formulations of Langlands.

A root-level core project because functoriality cuts across local, global, and geometric aspects of the program.

The abelian prototype from which Langlands reciprocity generalizes.

A celebrated application and prototype of global Langlands reciprocity.

A basic bridge between harmonic analysis on $p$-adic groups and the dual group.

A major solved family of cases of Langlands classification using endoscopy.

Develop the geometric formulation of local Langlands over $p$-adic fields and prove its equivalence with the classical representation-theoretic correspondence.

The foundational local correspondence for general linear groups over nonarchimedean local fields.

Establish the full local Langlands correspondence for arbitrary connected reductive groups over local fields, including packet structure, internal parameterization, and endoscopic character relations.

Extend the fully successful function-field theory for $GL_n$ to arbitrary reductive groups.

The central global reciprocity problem over number fields for general reductive groups.

A major realized portion of global reciprocity over number fields for $GL_n$.

The first full global Langlands theorem for higher rank groups.

The central conjectural transfer principle of the Langlands program.

Build the analytic and geometric machinery needed to prove functoriality in broad generality.

Classical and fundamental cases of functoriality.

A crucial endoscopic identity needed for stabilization and transfer.

The central categorical equivalence conjecture of geometric Langlands.

Develop and compare the major geometric realizations of the Langlands correspondence.

Foundational solved cases and constructions for geometric Langlands.

A geometric cornerstone underlying endoscopic and geometric Langlands methods.