Singularities of integrals: Homology, hyperfunctions and microlocal analysis

Main subject categories: • Singularities of integrals • Hyperfunctions • Integration on analytic sets • Integration on analytic spaces • Homology

Mathematics Subject Classification (2010): • 32C30 Integration on analytic sets and spaces, currents • 32A45 Hyperfunctions • 32A55 Singular integrals of functions in several complex variables • 32C81 Applications of analytic spaces to physics and other areas of science • 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) • 55N25 Homology with local coefficients, equivariant cohomology

Bringing together two fundamental texts from Frédéric Pham’s research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom’s isotopy theorems, Frédéric Pham’s foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis.

Providing a ‘must-have’ introduction to the singularities of integrals, a number of supplementary references also offer a convenient guide to the subjects covered.

This book will appeal to both mathematicians and physicists with an interest in the area of singularities of integrals.

Frédéric Pham, now retired, was Professor at the University of Nice. He has published several educational and research texts. His recent work concerns semi-classical analysis and resurgent functions.