Derived structures in the Langlands Correspondence

TL;DR

This paper surveys recent developments involving derived structures in the Langlands correspondence, including derived deformation rings, Hecke algebras, Hitchin stacks, and special cycles, emphasizing open problems for future research.

Contribution

It provides a comprehensive overview of how derived algebraic geometry is applied to the Langlands program, highlighting new structures and open questions.

Findings

Derived Galois deformation rings and Hecke algebras exhibit new properties.

Derived Hitchin stacks offer novel insights into geometric Langlands.

Identification of open problems guiding future research in the field.

Abstract

We survey several recent examples of derived structures emerging in connection with the Langlands correspondence. Cases studies include derived Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and derived special cycles. We also highlight some open problems that we expect to be important for future progress.