Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turrittin Theorem

When I first started learning about irregular perverse sheaves, I struggled to find a clear answer to the question: “what does an irregular perverse sheaf look like, in the simplest cases? What should be my intuition?” If I don’t know about the motivation from differential equations, how can I picture them? This is not yet… Continue reading Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turrittin Theorem

Deligne’s regular solution in dimension 1

In this post, I want to recall the elements of the regular Riemann-Hilbert Correspondence (but only in dimension 1, on our small disk around the origin). We’ll talk more about the category of Meromorphic Connections on with singularities at 0, and how they’re just a different way of phrasing linear ODEs whose solutions have singularities… Continue reading Deligne’s regular solution in dimension 1

Irregular Riemann-Hilbert Correspondence: introduction to the problem

One of the most successful bridges between analysis and algebraic geometry is the classical Riemann-Hilbert (R-H) correspondence between regular holonomic D-modules and perverse sheaves on complex manifolds, where  is the sheaf of differential operators with holomorphic coefficients (proved independently by Kashiwara and Mebkhout in 1984). This correspondence is a far-reaching generalization of Hilbert’s 21st Problem asking about the existence… Continue reading Irregular Riemann-Hilbert Correspondence: introduction to the problem