This book presents the monodromy group, underlining the unifying role it plays in a variety of theories and mathematical areas. In singularity theory and algebraic geometry, the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations, one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations, there appear the Ecalle-Voronin-Martinet-Ramis moduli. Moreover, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions.

The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. Readers will quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.

This second edition has been enlarged by several sections, presenting new results appeared since the first edition.

Contains quick introductions into singularity theory, analytic theory of ODE's, holomorphic foliations, Galois theory, and some parts of algebraic geometry Puts together all aspects of the monodromy group Gathers results scattered across multiple sources

Autorentext

Henryk odek is a Polish mathematician and a professor of mathematical sciences. He specializes in mathematical physics and differential equations.

Inhalt

Preface.- 1. Analytic Functions and Morse Theory.- 2. Normal Forms of Functions.- 3. Algebraic Topology of Manifolds.- 4. Topology and Monodromy of Functions.- 5. Integrals along Vanishing Cycles.- 6. Vector Fields and Abelian Integrals.- 7. Hodge Structures and Period Map.- 8. Linear Differential Systems.- 9. Holomorphic Foliations. Local Theory.- 10. Holomorphic Foliations. Global Aspects.- 11. The Galois Theory.- 12. Hypergeometric Functions.- Bibliography.- Index.