Orlando Neto

A Microlocal Riemann–Hilbert Correspondence

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic D-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.

Singular regular differential equations and eigenvalues of products of matrices

. We survey some recent results on linear representations of free groups motivated by problems on linear differential equations and we generalize a theorem of Simpson on necessary and sufficient conditions for the existence of linear representations of finitely generated free groups with prescribed similarity classes.

Trees, band monoids and formal languages

We show that a relatively free band monoid generated by a set A can be understood as a set of maps from a certain binary tree into the boolean algebra (A). We describe the atoms of the boolean algebras of the varieties of languages corresponding to the pseudovarieties of band monoids.

Holonomic systems with solutions ramified along a cusp

Abstract We classify the holonomic systems of (micro) differential equations of multiplicity one along a singular Lagrangian irreducible variety contained in an involutive submanifold of maximal codimension. We show that their solutions are related to k F k −1 hypergeometric functions on the Riemann sphere. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 171–176.

On the recognition and rigidity problems for sums of matrices

Abstract Let S 0 ,…, S k be k +1 similarity classes of matrices over an algebraically closed field. This paper gives partial answers to the problems of existence ( recognition ) and uniqueness modulo simultaneous similarity ( rigidity ) of the solutions of the equation A k = A 0 +⋯+ A k −1 , with A i ∈ S i , 0⩽ i ⩽ k .

Limits of tangents of quasi-ordinary hypersurfaces

We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .

Equisingularity and Lagrangean Curves

The most natural candidates to the definition of equisingularity of germs of complex space curves are discussed, showing that they coincide in the case of Lagrangean curves (projectivizations of conic Lagrangean varieties of a symplectic manifold of dimension four).

Systems of meromorphic microdifferential equations

We introduce the notion of system of meromorphic microdifferential equations. We use it to prove a desingularization theorem for systems of microdifferential equations.