Table of Contents

1 Symplectic geometry.- 1.1 Symplectic manifolds.- 1.2 Submanifolds of symplectic manifolds.- 1.3 Lagrangian manifolds, fibrations, mappings, and singularities.- 2 Applications of the theory of Lagrangian singularities.- 2.1 Oscillatory integrals.- 2.2 Lattice points.- 2.3 Perestroikas of caustics.- 2.4 Perestroikas of optical caustics.- 2.5 Shock wave singularities and perestroikas of Maxwell sets.- 3 Contact geometry.- 3.1 Wave fronts.- 3.2 Singularities of fronts.- 3.3 Perestroikas of fronts.- 4 Convolution of invariants, and period maps.- 4.1 Vector fields tangent to fronts.- 4.2 Linearised convolution of invariants.- 4.3 Period maps.- 4.4 Intersection forms of period maps.- 4.5 Poisson structures.- 4.6 Principal period maps.- 5 Lagrangian and Legendre topology.- 5.1 Lagrangian and Legendre cobordism.- 5.2 Lagrangian and Legendre characteristic classes.- 5.3 Topology of complex discriminants.- 5.4 Functions with mild singularities.- 5.5 Global properties of singularities.- 5.6 Topology of Lagrangian inclusions.- 6 Projections of surfaces, and singularities of apparent contours.- 6.1 Singularities of projections from a surface to the plane.- 6.2 Singularities of projections of complete intersections.- 6.3 Geometry of bifurcation diagrams.- 7 Obstacle problem.- 7.1 Asymptotic rays in symplectic geometry.- 7.2 Contact geometry of pairs of hypersurfaces.- 7.3 Unfurled swallowtails.- 7.4 Symplectic triads.- 7.5 Contact triads.- 7.6 Hypericosahedral singularity.- 7.7 Normal forms of singularities in the obstacle problem.- 8 Transformation of waves defined by hyperbolic variational principles.- 8.1 Hyperbolic systems and their light hypersurfaces.- 8.2 Singularities of light hypersurfaces of variational systems.- 8.3 Contact normal forms of singularities of quadratic cones.- 8.4 Singularities of ray systems and wave fronts at nonstrict hyperbolic points.