Algebraic Approach to Differential Equations

The following sections are included:

• PREFACE

• ACKNOWLEDGMENTS

• CONTENTS

The following sections are included:

• Introduction

• Cauchy Theorem

• Sheaves of Holomorphic Functions

• Sheaf Version of Cauchy Theorem

• Local Monodromy

• Fuchs Theory

• Index of Differential Operators at Singular Points

• Division Tools

• Generalized Solutions

• Holonomic D-Modules

• Regular D-Modules

• A Local Version of the Riemann-Hilbert Correspondence in One Variable (in Collaboration with F. Gudiel Rodríguez)

• D-Modules on a Riemann Surface

• References

We develop some basic results on modules over the Weyl algebra including the existence of the Bernstein polynomial and the relationship between logarithmic modules and the so-called Logarithmic Comparison Theorem. We use computer algebra system Macaulay 2 to explicitly compute some invariants arising in the whole subject.

The following sections are included:

• Introduction

• Whitney Conditions

• Stratifications

• Constructible Sheaves

• Whitney Stratifications

• Milnor Fibrations

• Local Constructible Sheaves and Whitney Conditions

• Neighbouring Cycles

• Constructible Complexes

• Vanishing Cycles

• Holonomic -Modules

• References

The paper is based on a course given in 2007 at an ICTP school in Alexandria, Egypt. It aims at introducing young scientists to methods to calculate asymptotic developpments of singular integrals.

This is a short presentation of some relations between the arrangements of hyperplanes and the theory of hypergeometric functions. All of this paper is coming from the illuminating monograph written by P. Orlik and H. Terao⁹ where all the detailed proofs could be found. Moreover, we focus our interest on the part of this theory which consists of the computation of the cohomology groups, part of the hypergeometric pairing. We mainly consider the non-resonant weights and we just give some insight on the resonant case.

The following sections are included:

• Introduction to Functional Equations

• On the existence of a functional equation in the polynomial case

• Bernstein-Sato polynomials and analytic continuation of f^s

• Quasi-homogeneous and semi-quasi-homogeneous isolated singularities

• Proof of the existence of the b-function for a an analytic germ

• Rationality of the zeroes over any algebraically closed field of characteristic zero

• Introduction to analytic differential modules. Abridged version

• Complements

• References

Given a family of algebraic groups G → S depending on a parameter S, one studies systems of partial differential equations with variables in S and values in G, which are compatible with the group structure. Two cases are especially studied: the commutative case, and the simple case.