Many important functions of mathematical physics have integral representa-
tions, i.e. are defined by integrals depending on parameters. Such functions include,
in particular, the fundamental solutions of a majority of classical partial differential
equations, Newton-Coulomb potentials, integral Fourier transforms, initial data of
inverse tomography problems, hypergeometric functions, Feynman integrals, etc.
The general construction of these integral representations is as follows. Suppose we
have an analytic fiber bundle E — T, an exterior differential form uo on E, whose
restrictions on the fibers are closed, and a family of integration cycles in these
fibers, parametrized by the corresponding points of the base T and depending con-
tinuously on these points. Then the integral UJ along these cycles is a function on
Analytic and qualitative properties of such functions depend on the monodromy
of these cycles, i.e. on the natural action of the fundamental group of the base in the
homology groups of the fibers: this action defines the ramification of the analytic
continuation of our integral function.
The study of this action (which is a purely topological problem) allows us to
answer questions on the analytic behavior of the integral function, in particular
whether this function is single-valued or at least algebraic, what are the singular
points of this function, and (partially) what is its asymptotics close to these points.
Ramification of integral functions arising in different problems is described
by different (but having some common features) versions of the Picard-Lefschetz
theory. Our book contains a list of such versions, including the classical local mon-
odromy theory of singularities of functions and complete intersections, F. Pham's
generalized Picard-Lefschetz formulas, stratified version of the theory (studying in
particular the monodromy of homology groups of hyperplane sections of singular
varieties), and also twisted versions of all these theories (related to integrals of
Using them, we study four famous classes of functions:
— volume functions arising in the Archimedes-Newton problem on integrable
— Newton-Coulomb potentials,
— fundamental solutions of hyperbolic partial differential equations (studied,
in particular, in the lacuna theory of Hadamard-Petrovskh-Atiyah-Bott-
— multidimensional (Gelfand-Aomoto) hypergeometric functions generalizing
the Gaufi hypergeometric integral.
Some of main results described in the book are as follows.
1. Newton's theorem on the algebraic nonintegrability of plane ovals (stating
that the function on the space of lines in R2, associating with any line