CRITICAL POINTS OF SMOOTH FUNCTIONS AND THEIR NORMAL FORMS

This paper contains a survey of research on critical points of smooth functions and their bifurcations. We indicate applications to the theory of Lagrangian singularities (caustics), Legendre singularities (wave fronts) and the asymptotic behaviour of oscillatory integrals (the stationary phase method). We describe the connections with the theories of groups generated by reflections, automorphic forms, and degenerations of elliptic curves. We give proofs of the theorems on the classification of critical points with at most one modulus, and also a list of all singularities with at most two moduli. The proofs of the classification theorems are based on a geometric technique associated with Newton polygons, on the study of the roots of certain Lie algebras resembling the Enriques-Demazure technique of fans, and on spectral sequences that are constructed with respect to quasihomogeneous filtrations of the Koszul complex defined by the partial derivatives of a function.

[1]  John Guckenheimer,et al.  Bifurcation and Catastrophe††AMS (MOS) 1970 SUBJECT CLASSIFICATION: 58F99.‡‡Research supported by National Science Foundation (GP-7952X2) at the Institute for Advanced Study, Princeton, New Jersey. , 1973 .

[2]  Alan Durfee.  Fibered knots and algebraic singularities , 1974 .

[3]  A. Gabrielov.  Bifurcations, Dynkin diagrams, and modality of isolated singularities , 1974 .

[4]  I. Dolgachev.  Automorphic forms and quasihomogeneous singularities , 1975 .

[5]  A N Varčenko,et al.  THEOREMS ON THE TOPOLOGICAL EQUISINGULARITY OF FAMILIES OF ALGEBRAIC VARIETIES AND FAMILIES OF POLYNOMIAL MAPPINGS , 1972 .

[6]  M. Golubitsky.  Contact equivalence for Lagrangian submanifolds , 1975 .

[7]  J. Guckenheimer.  Catastrophes and partial differential equations , 1973 .

[8]  René Thom,et al.  Structural stability and morphogenesis , 1977, Pattern Recognit..

[9]  Claus Michael Ringel,et al.  Indecomposable Representations of Graphs and Algebras , 1976 .

[10]  N. A'campo.  La fonction zêta d'une monodromie , 1975 .

[11]  I. Gel'fand,et al.  FREE MODULAR LATTICES AND THEIR REPRESENTATIONS , 1974 .

[12]  J. Wagoner,et al.  Pseudo-isotopies of compact manifolds , 1973 .

[13]  R. Thom,et al.  Un résultat sur la monodromie , 1971 .

[14]  J. Milnor.  Singular points of complex hypersurfaces , 1968 .

[15]  J. Leray,et al.  Théorie des perturbations et méthodes asymptotiques , 1972 .

[16]  Eduard Looijenga,et al.  A period mapping for certain semi-universal deformations , 1975 .

[17]  P. Orlik,et al.  Isolated Singularities of Algebraic Surfaces with C ∗ Action , 1971 .

[18]  Kyoji Saito,et al.  Quasihomogene isolierte Singularitäten von Hyperflächen , 1971 .

[19]  V. Arnold.  SINGULARITIES OF SMOOTH MAPPINGS , 1968 .

[20]  J. Guckenheimer,et al.  Isochrons and phaseless sets , 1975, Journal of mathematical biology.

[21]  V. Arnold.  LECTURES ON BIFURCATIONS IN VERSAL FAMILIES , 1972 .

[22]  L. Nazarova.  REPRESENTATIONS OF QUIVERS OF INFINITE TYPE , 1973 .

[23]  Gordon S Wassermann,et al.  Stability of Unfoldings , 1974 .

[24]  Le Dung Trang,et al.  The invariance of Milnor's number implies the invariance of the topological type , 1976 .

[25]  Claus Michael Ringel,et al.  Representations of graphs and algebras , 1974 .

[26]  V. Arnold.  REMARKS ON THE STATIONARY PHASE METHOD AND COXETER NUMBERS , 1973 .

[27]  Stability of caustics , 1975 .

[28]  M. V. Berry,et al.  Waves and Thom's theorem , 1976 .

[29]  Klaus Lamotke.  Die Homologie isolierter Singularitäten , 1975 .

[30]  K. Jänich.  Caustics and catastrophes , 1974 .

[31]  A. Gabrielov.  Intersection matrices for certain singularities , 1973 .

[32]  J. Cerf.  La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie , 1970 .

[33]  John Guckenheimer,et al.  Shocks and rarefactions in two space dimensions , 1975 .

[34]  Robert Steinberg,et al.  Regular elements of semisimple algebraic groups , 1965 .

[35]  V. I. Arnol'd,et al.  Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities , 1972 .

[36]  Felix Klein,et al.  Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert , 1967 .

[37]  Auguste Fresnel.  Memoire sur la diffraction de la lumiere , 1819 .

[38]  Catastrophe theory in biology , 1975 .

[39]  J. Tougeron,et al.  Idéaux et fonctions différentiables. I , 1968 .

[40]  N. A'campo.  Le nombre de Lefschetz d'une monodromie , 1973 .

[41]  E. Brieskorn.  Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe , 1971 .

[42]  Johannes J. Duistermaat,et al.  Oscillatory integrals, lagrange immersions and unfolding of singularities , 1974 .

[43]  Graeme Segal,et al.  Configuration-spaces and iterated loop-spaces , 1973 .

[44]  A. Samoilenko.  The equivalence of a smooth function to a Taylor polynomial in the neighborhood of a finite-type critical point , 1968 .

[45]  S. Gusein-Zade.  Characteristic polynomial of classical monodromy for singular series , 1976 .

[46]  A. Gabrielov,et al.  Description of deformations with constant milnor number for homogeneous functions , 1975 .

[47]  E. Looijenga.  The complement of the bifurcation variety of a simple singularity , 1974 .

[48]  Egbert Brieskorn,et al.  Beispiele zur Differentialtopologie von Singularitäten , 1966 .

[49]  J. Guckenheimer.  Solving a single conservation law , 1975 .

[50]  Vladimir I. Arnold,et al.  Wave front evolution and equivariant Morse lemma , 1976 .

[51]  V. Arnold,et al.  Galina Nikolaevna Tyurina (obituary) , 1971 .

[52]  E. Brieskorn.  Rationale Singularitäten komplexer Flächen , 1968 .

[53]  B. Malgrange,et al.  Le polynome de Bernstein d’une singularite isolee , 1975 .

[54]  Gordon Wassermann,et al.  Stability of unfoldings in space and time , 1975 .

[55]  J. Steenbrink.  Limits of Hodge structures , 1976 .

[56]  W. Stoll.  The multiplicity of a holomorphic map , 1966 .

[57]  R. Thom.  Stabilité structurelle et morphogenèse , 1974 .

[58]  P. Deligne,et al.  Les immeubles des groupes de tresses généralisés , 1972 .

[59]  A. Hatcher.  Parametrized $h$-cobordism theory , 1973 .

[60]  S. Kaliman.  Holomorphic universal covering spaces of polynomials without multiple roots , 1975 .

[61]  Egbert Brieskorn,et al.  Artin-Gruppen und Coxeter-Gruppen , 1972 .

[62]  Michael Atiyah,et al.  Resolution of Singularities and Division of Distributions , 1970 .

[63]  N. A'campo.  Le groupe de monodromie du déploiement des singularités isolées de courbes planes I , 1975 .

[64]  D. B. Fuks.  Quillenization and bordism , 1974 .

[65]  L. Tráng,et al.  Un théorème de Zariski du type de Lefschetz , 1973 .

[66]  Michel Demazure,et al.  Sous-groupes algébriques de rang maximum du groupe de Cremona , 1970 .

[67]  P. Gabriel.  Unzerlegbare Darstellungen I , 1972 .

[68]  L. Hörmander.  Fourier integral operators. I , 1995 .

[69]  M. Artin,et al.  On the solutions of analytic equations , 1968 .

[70]  G. N. Tjurina.  THE TOPOLOGICAL PROPERTIES OF ISOLATED SINGULARITIES OF COMPLEX SPACES OF CODIMENSION ONE , 1968 .

[71]  R. Thom.  The bifurcation subset of a space of maps , 1971 .

[72]  J. Guckenheimer.  Caustics and non-degenerate Hamiltonians☆ , 1974 .

[73]  J. Milnor,et al.  Isolated singularities defined by weighted homogeneous polynomials , 1970 .

[74]  L. Nazarova,et al.  Representations of partially ordered sets , 1975 .

[75]  F. Pham.  Formules de Picard-Lefschetz généralisées et ramification des intégrales , 1965 .

[76]  E. C. Zeeman,et al.  Levels of Structure in Catastrophe Theory Illustrated by Applications in the Social and Biological Sciences , 1975 .

[77]  W. Hodge.  The Isolated Singularities of an Algebraic Surface , 1930 .

[78]  C. Ringel,et al.  On algebras of finite representation type , 1975 .

[79]  N. A'campo.  Sur la monodromie des singularités isolées d'hypersurfaces complexes , 1973 .

[80]  Egbert Brieskorn,et al.  Die monodromie der isolierten singularitäten von hyperflächen , 1970 .

[81]  S. Gusein-Zade.  Dynkin diagrams for singularities of functions of two variables , 1974 .

[82]  Kyoji Saito.  Einfach-elliptische Singularitäten , 1974 .

[83]  V. I. Arnol’d.  Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds , 1972 .

[84]  D. Siersma.  The singularities of C∞-functions of right-codimension smaller or equal than eight , 1973 .

[85]  G. N. Tyurina.  Resolution of singularities of plane deformations of double rational points , 1970 .

[86]  I. Gelfand,et al.  COXETER FUNCTORS AND GABRIEL'S THEOREM , 1973 .

[87]  V. Arnold.  On some topological invariants of algebraic functions , 1970 .

[88]  R. Hartshorne.  Residues And Duality , 1966 .

[89]  Oscar Zariski,et al.  A Theorem on the Poincare Group of an Algebraic Hypersurface , 1937 .

[90]  S. Gusein-Zade.  Intersection matrices for certain singularities of functions of two variables , 1974 .