I The topological structure of isolated critical points of functions. - 1 Elements of the theory of Picard-Lefschetz. - 2 The topology of the non-singular level set and the variation operator of a singularity. - 3 The bifurcation sets and the monodromy group of a singularity. - 4 The intersection matrices of singularities of functions of two variables. - 5 The intersection forms of boundary singularities and the topology of complete intersections. - II Oscillatory integrals. - 6 Discussion of results. - 7 Elementary integrals and the resolution of singularities of the phase. - 8 Asymptotics and Newton polyhedra. - 9 The singular index examples. - III Integrals of holomorphic forms over vanishing cycles. - 10 The simplest properties of the integrals. - 11 Complex oscillatory integrals. - 12 Integrals and differential equations. - 13 The coefficients of series expansions of integrals the weighted and Hodge filtrations and the spectrum of a critical point. - 14 The mixed Hodge structure of an isolated critical point of a holomorphic function. - 15 The period map and the intersection form. - References. Language: English