Eduardo Corel | Searcharxiv

arXiv2012

Gérard-Levelt membranes

We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical Gérard-Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat-Tits building, that we call the Gérard-Levelt membrane. This provides a way to compute the true Poincaré rank, but also the Katz rank of a meromorphic connection without having to perform gauge transforms nor ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila's method for Bergman fans to the case of the tight-span of a valuated matroid.

math.CA↗

arXiv2010

Stable Flags and the Riemann-Hilbert Problem

We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise logarithmic modifications of the classical Röhrl-Deligne vector bundle. We show that the solutions of the Riemann-Hilbert problem are in bijection with some families of local filtrations which are stable under the prescribed monodromy maps. We introduce the notion of Birkhoff-Grothendieck trivialisation, and show that its computation corresponds to geodesic paths in some local affine Bruhat-Tits building. We use this to compute how the type of a bundle changes under stalk modifications, and give several corresponding algorithmic procedures.

math.CA↗