119844353
REMARKS ON THE STATIONARY PHASE METHOD AND COXETER NUMBERS
We study integrals of rapidly oscillating functions. Such integrals tend to zero when the length of the oscillation wave tends to zero through wave fronts of constant form. The asymptotic decrease of the integral is determined by the character of the critical points of the function describing the front. If all of these critical points are non-degenerate (Morse), then the integral tends to zero like the wave length raised to the power of half the dimension of the space, and indeed this is the asymptotic behaviour of the integral for functions in general position.However, if the integral depends on additional parameters, then for certain caustic parameter values there arise non-Morse critical points and the integral decreases slowly. The investigation of the asymptotic behaviour of the integral of an oscillating function in caustic cases can be regarded as a generalization of the theory of Airy functions; it is closely connected with Artin's braid theory, and the answer in the case of few parameters is expressed in terms of the Coxeter numbers of the Weyl groups of the series A, D, E, and in the case of many parameters it is expressed in terms of generalizations of them.
122136489
Spaces of functions with moderate singularities
121836338
Wave front evolution and equivariant Morse lemma
120329630
Weak asymptotics for the numbers of solutions of diophantine problems
122633217
Poisson structures on the plane and other powers of volume forms
The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.
122507359
The Theory of Singularities and its Applications
Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.
118067094
Singularities of smooth mappings with additional structures : collected papers