The author systematically develops an approach based on singularity theory to the investigation of various special functions admitting an integral representation and occurring in physics, integral geometry, partial differential equations, etc. His book is devoted to the study of four important classes of functions: the volume functions, the Newton-Coulomb potentials, the Green functions of hyperbolic equations, and the multidimensional hypergeometric functions of Gelfand-Aomoto types.
The book consists of the preface, introduction, eight chapters and bibliography including 189 references. In the preface main results described in the book are listed. Next is the introduction written in a didactic style and containing an explanation of key ideas and basic results with a series of nonformal comments and remarks.
In the first two chapters the classical Picard-Lefschetz theory and its relations with the singularity theory of functions are discussed at great length. In particular, the author covers the following topics: critical points and values, Milnor fibre, monodromy and variation operators, Gauss-Manin connexion, Picard-Lefschetz formula, Dynkin diagrams, \({\mathcal R}\)-classification of real smooth and complex isolated hypersurface singularities, stratifications, intersection homology theory, etc. These chapters are particularly apt for those who want to enter topological aspects of singularity theory.
The third and seventh chapters deal with properties of the volume function whose notion (in the planar case) goes back to Isaac Newton. In the third chapter the non-algebraicity is proved of the volume function defined by any convex compact hypersurface in the even-dimensional real space. In the odd-dimensional case there are many topological obstructions for such a function to be algebraic. This leads to the conjecture that the only example of algebraic volume function is defined by the ellipsoids in \({\mathbb R}^{2n-1}\) (the so-called Archimedes’ example). In chapter VII the author studies the Newton-Coulomb potential of hyperbolic algebraic surfaces in \({\mathbb R}^n.\) Three centuries ago Newton found that the potential of a sphere is constant inside the sphere. One hundred twenty two years later, J. Ivory proved the analogous statement for ellipsoids. Recently, natural generalizations of these classical results to the case of arbitrary hyperbolic hypersurfaces in \({\mathbb R}^n\) have been obtained by V. I. Arnold and V. A. Vassiliev [Notices Am. Math. Soc. 36, No. 9, 1148–1154 (1989; Zbl 0693.01005)]. Against such a historical background the author examines the case of a smooth hyperbolic hypersurface in \({\mathbb R}^n\) and analyses the behavior of Newtonian potentials outside the hyperbolicity domain. As a result, he describes all cases when such potentials of general hyperbolic hypersurfaces are algebraic. Furthermore conditions under which the potential is algebraic outside the hypersurface are determined and investigated; an approach to the study of the odd-dimensional case is also discussed. The main idea of the author’s observations is to use an integral representation of the potential function, easy considerations from the monodromy theory of complete intersections, and properties of homology groups with coefficients in a local system.
Chapters IV and V are devoted to the study of the lacuna problem for hyperbolic differential operators with constant coefficients. First the author recalls the classification of the singular points of wave fronts for hyperbolic operators with constant coefficients, the description of local lacunae close to nonsingular points of fronts (following A. M. Davydova and V. A. Borovikov) and to the singularities \(A_2\) and \(A_3\) [following L. Gårding, Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl., 53–68 (1977; Zbl 0369.35062)]. Of course, these results are reformulated in terms of singularity theory. Then the local Petrovskii cycles of strictly hyperbolic operators are also expressed in such a manner and the equivalence of local regularity of fundamental solutions of hyperbolic PDEs and the topological Petrovskii-Atiyah-Bott-Gårding condition is proved. Finally, a combinatorial algorithm which enumerates local lacunae close to all simple (and many nonsimple) singular points of wave fronts is described in detail.
Chapter VI is devoted to the investigation of homology of local systems, twisted monodromy theory and regularization problem of improper integration cycles. It contains, among other things, a description of twisted vanishing homology of complete intersection singularities and extensions of results from the second chapter that are mainly based on stratified Picard-Lefschetz theory with twisted coefficients.
The final chapter is concerned with the theory of multidimensional hypergeometric functions in the sense of Gelfand and Aomoto; the ramification, singularities, resonance and integral representations of such functions are also investigated. In particular, the author proves a well-known theorem on the analytic continuation of the multidimensional hypergeometric functions and discusses related problems; under certain condition he also obtains a detail description of the case of real plane arrangements [V. A. Vassiliev, I. M. Gelfand and A. V. Zelevinskij, Funkts. Anal. Prilozh. 21, No. 1, 23–38 (1987; Zbl 0614.33008)], and so on.
It should be remarked that the book under review can be considered as an expanded and revised version of [V. A. Vassiliev, Ramified integrals, singularities and lacunas. Mathematics and its Application. 315 Kluwer (1995; Zbl 0935.32026)]; a significant part of new materials is based on recent results and ideas of the author and his collaborators. The book contains many clear examples, comments, remarks, open questions and problems illustrated by lot of tables, pictures, and diagrams as well as important applications with instructive references and historical background. With no doubt this book is comprehensible, interesting and useful for graduate students; the variety of topics covered makes it also highly valuable for researchers, lecturers, and practicians working in either of the above mentioned fields of mathematics and its applications.