The 17 invited research articles in this volume, all written by leading experts in their respective fields, are dedicated to the great French mathematician Jean Leray. A wide range of topics with significant new results---detailed proofs---are presented in the areas of partial differential equations, complex analysis, and mathematical physics. Key subjects are: * Treated from the mathematical physics viewpoint: nonlinear stability of an expanding universe, the compressible Euler equation, spin groups and the Leray--Maslov index, * Linked to the Cauchy problem: an intermediate case between effective hyperbolicity and the Levi condition, global Cauchy--Kowalewski theorem in some Gevrey classes, the analytic continuation of the solution, necessary conditions for hyperbolic systems, well posedness in the Gevrey class, uniformly diagonalizable systems and reduced dimension, and monodromy of ramified Cauchy problem. Additional articles examine results on: * Local solvability for a system of partial differential operators, * The hypoellipticity of second order operators, * Differential forms and Hodge theory on analytic spaces, * Subelliptic operators and sub- Riemannian geometry. Contributors: V. Ancona, R. Beals, A. Bove, R. Camales, Y. Choquet- Bruhat, F. Colombini, M. De Gosson, S. De Gosson, M. Di Flaviano, B. Gaveau, D. Gourdin, P. Greiner, Y. Hamada, K. Kajitani, M. Mechab, K. Mizohata, V. Moncrief, N. Nakazawa, T. Nishitani, Y. Ohya, T. Okaji, S. Ouchi, S. Spagnolo, J. Vaillant, C. Wagschal, S. Wakabayashi The book is suitable as a reference text for graduate students and active researchers.

Contenuti

Preface * Differential Forms, Cycles and Hodge Theory on Complex Analytic Spaces * On Exact Solutions of Linear PDEs * Necessary Conditions for Hyperbolic Systems * Monodromy of the Ramified Cauchy Problem * Non Linear Stability of an Expanding Universe with S^1 Isometry Group * On the Cauchy Problem for a Weakly Hyperbolic Operator: An Intermediate Case Between Effective Hyperbolicity and Levi Condition * Symplectic Path Intersections and the Leray Index * A Global Cauchy-Kowalewski Theorem in Some Gevrey Classes * Subriemannian Geometry and Subelliptic PDEs * On the Analytic Continuation of the Solution of the Cauchy Problem * Strong Gevrey Solvability for a System of Linear Partial Differential Equations * Spherically Symmetric Solutions of the Compressible Euler Equation * Hyperbolic Cauchy Problem Well Posed in the Class of Gevrey * Absence of Eigenvalues of Dirac Type Operators * The Behaviors of Singular Solutions of Partial Differential Equations in Some Class in the Complex Domain * Syst�mes Uniform�ment Diagonalisables, Dimension R�duite et Sym�trie II * On Hypoellipticity of the Operator exp [-|x_1|^{-\sigma }]D_1^2+x_1^4 D_2^2+1

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