|May 08, 2014
The original Riemann-Hilbert problem is to construct a liner ordinary differential equation with regular singularities whose solutions have a given monodromy.
Nowadays, it is formulated as a categorical equivalence of the category of regular holonomic D-modules and the category of perverse sheaves.
It is a long standing problem to describe liner ordinary differential equations with irregular singularities in geometric terms.
Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374).
In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line.
Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.