M. van der Put

Algebraic subgroups of GL2(C)

Abstract In this note we classify, up to conjugation, all algebraic subgroups of GL 2 (ℂ).

A Riemann-Hilbert approach to Painleve IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ℙ1 are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.

Formal theory for differential-difference operators

Differential-difference operators are linear operators involving both d/dz and the shift .The aim is to give a formal classification and to provide solutions for these equations. Differential-difference operators can be considered as formal differential operators of infinite order. For the latter one studies Newton polygons, factorizations, solutions and developes a theory of symbolic solutions. This theory applied to differential-difference operators seems in many case adequate. In other cases, one cannot produce enough symbolic solutions. Independent from differential operators of infinite order, certain systems of differential-difference are treated. Here the theory seems complete. Many examples illustrate the theory.

Invariants of analytic local rings

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