The planned research introduces a new formulation of the categorical Langlands correspondence. It replaces the `naive' version of the Langlands correspondence, which is known to specialists to be inconsistent. The principal investigator proposes to study the new statement and verify that it passes natural consistency checks. He then plans to verify the new formulation, starting with some special cases. The investigator also proposes to study the compactified Jacobians of curves with planar singularities and show that they are self-dual. In his earlier work, the principal investigator established the autoduality in the case of reduced irreducible curves; he plans to extend his results to curves that may be reducible and non-reduced.
The Langlands program started as a series of conjectures proposed by Robert Langlands beginning in 1967. From its origins in number theory and representation theory, it developed into a framework connecting and influencing many areas of mathematics. In particular, the geometric Langlands program is formulated in terms of algebraic geometry, but can also be interpreted in physical terms as a duality in quantum field theory. This project introduces new geometric ideas into the Langlands program that should resolve some long-standing issues.
