Irreducible flat SL(2,R)-connections on the trivial holomorphic bundle

Monodromies of Algebraic Connections on the Trivial Bundle

In this note, we study monodromies of algebraic connections on the trivial vector bundle. We prove that on a smooth complex affine curve, any monodromy arises as the underlying local system of an algebraic connection on the trivial bundle. We give a generalization of this for rank one monodromies in higher dimension. Monodromies des connexions algébriques sur le fibré trivial Résumé. Dans cette...

Trivial Connections on Discrete Surfaces

This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections by solving a single linear system built from standard operators. The solution can be used to design rotationally symmetric direction fields with user-specified singulariti...

Irreducible Cyclic Presentations of the Trivial Group

Lectures on meromorphic flat connections

These notes form an extended version of a minicourse delivered in Université de Montréal (June 2002) within the framework of a NATO workshop “Normal Forms, Bifurcations and Finiteness Problems in Differential Equations”. The focus is on Poincaré–Dulac theory of “Fuchsian” (logarithmic) singularities of integrable systems, with applications to problems on zeros of Abelian integrals in view.

Holomorphic Connections on Filtered Bundles over Curves

Let X be a compact connected Riemann surface and EP a holomorphic principal P–bundle over X , where P is a parabolic subgroup of a complex reductive affine algebraic group G. If the Levi bundle associated to EP admits a holomorphic connection, and the reduction EP ⊂ EP × P G is rigid, we prove that EP admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition...