"...A nice feature of the book [is] that at various points the authors provide examples, or rather counterexamples, that clearly show what can go wrong...This is a nicely-written book [that] studies algebraic differential modules in several variables."

--Mathematical Reviews

1 Regularity in several variables.- Geometric models of divisorially valued function fields.- Logarithmic differential operators.- Connections regular along a divisor.- Extensions with logarithmic poles.- Regular connections: the global case.- Exponents.- Appendix A: A letter of Ph. Robba (Nov. 2, 1984).- Appendix B: Models and log schemes.- 2 Irregularity in several variables.- Spectral norms.- The generalized Poincar-Katz rank of irregularity.- Some consequences of the Turrittin-Levelt-Hukuhara theorem.- Newton polygons.- Stratification of the singular locus by Newton polygons.- Formal decomposition of an integrable connection at a singular divisor.- Cyclic vectors, indicial polynomials and tubular neighborhoods.- 3 Direct images (the Gauss-Manin connection).- Elementary fibrations.- Review of connections and De Rham cohomology.- Dissage.- Generic finiteness of direct images.- Generic base change for direct images.- Coherence of the cokernel of a regular connection.- Regularity and exponents of the cokernel of a regular connection.- Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case).- Appendix C: Berthelot's comparison theorem on OXDX-linear duals.- Appendix D: Introduction to Dwork's algebraic dual theory.- 4 Complex and p-adic comparison theorems.- Review of analytic connections and De Rham cohomology.- Abstract comparison criteria.- Comparison theorem for algebraic vs.complex-analytic cohomology.- Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients).- Rigid-analytic comparison theorem in relative dimension one.- Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients).- The relative non-archimedean Turrittin theorem.- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach.- References.