By Frédéric Pham

Bringing jointly primary texts from Frédéric Pham’s learn on singular integrals, the 1st a part of this e-book makes a speciality of topological and geometrical points whereas the second one explains the analytic strategy. utilizing notions constructed via J. Leray within the calculus of residues in numerous variables and R. Thom’s isotopy theorems, Frédéric Pham’s foundational examine of the singularities of integrals lies on the interface among research and algebraic geometry, culminating within the Picard-Lefschetz formulae. those mathematical constructions, enriched by means of the paintings of Nilsson, are then approached utilizing tools from the speculation of differential equations and generalized from the perspective of hyperfunction thought and microlocal analysis.

Providing a ‘must-have’ advent to the singularities of integrals, a couple of supplementary references additionally provide a handy consultant to the topics covered.

This publication will attract either mathematicians and physicists with an curiosity within the sector of singularities of integrals.

Frédéric Pham, now retired, was once Professor on the college of great. He has released numerous academic and examine texts. His contemporary paintings matters semi-classical research and resurgent functions.

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**Additional resources for Singularities of integrals: Homology, hyperfunctions and microlocal analysis**

**Example text**

Zr near a, such that S = {z : z1 = · · · = zr = 0}. The ideal spanned by z1 , z2 , . . , zr at the point a (the origin of the coordinates) is of rank r, and is obviously equal to its own radical: it is therefore ia (S). 6 Irreducibility The following deﬁnitions can just as well be applied locally (to germs of analytic sets) as globally (to analytic sets). An analytic set S is said to be irreducible if it cannot be written as the union of two analytic sets which are distinct from it. Otherwise, it is said to be reducible.

We can then pose the following algebraic problem: since C ∗ (X) is the Z-dual of C∗ (X) by | , can we deduce that H ∗ (X) is the Z-dual of H∗ (X) by | ? The answer is yes, modulo torsion. If, instead of taking coeﬃcients in Z, we took them in a ﬁeld, then we avoid any torsion problems and H ∗ (X) is the vector space which is dual to H∗ (X). 1 becomes particularly interesting in the light of de Rham’s theorem. This states that we can replace C ∗ (X) by Ω(X), δ by d, and | by integration in the deﬁnition of cohomology H ∗ (X) with coeﬃcients in R or C (the formula δϕ | γ = ϕ | ∂γ then becomes Stokes’ formula).

1, so that we can deﬁne its cohomology: H ∗ (X, A) = Z ∗ (X, A)/B ∗ (X, A) [= Φ(X, A)/dΩ(X, A)], B ∗ (X, A) = Im δ. 4) that if A is a closed diﬀerentiable submanifold of X, the restriction homomorphism of diﬀerential forms is surjective. 4 De Rham duality 39 a way that ψ exists). Then δψ deﬁnes a cocycle on X, whose restriction to A is zero: δψ|A = i∗ δψ = δi∗ ψ = δϕ = 0, and is therefore a relative cocycle on (X, A). One checks that its relative cohomology class, denoted δ ∗ hp ∈ H p+1 (X, A), only depends on the original class hp .