Introduction Sturm theory extends the Morse inequality (minorating the number of critical points of functions on a circle) to the higher derivatives. The Legendrian Morse theory (created by Yu. V. Chekanov in 1986) provides the Morse inequality for the multivalued functions (corresponding to the unknoted Legendrian submanifolds of the space of 1-jets of functions). It is a generalization of the Lagrangian intersection theory due to Conley, Zehnder, Chaperon, Floer, Sikorav, Laudenbach, Hofer, Gromov, and others. Below an attempt to extend the Legendrian Morse theory to higher derivatives is presented. It extends the Legendrian Morse theory in the same sense in which the ordinary Sturm theory extends Morse theory. Sturm theory implies the existence of at least four flattening points on every curve in projective 3-space having a convex projection. The present paper results imply, for instance, the preservation of at least four flattening points under (not necessary small) deform

Let F be a smooth function of the variables (x, y, p) e IR 3 and consider the first

We show that the existence of algebraic forms of exactly-solvable A−B− C −D and G2, F4 Olshanetsky-Perelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.

Abstract. The hyperbolic Schwarz map is defined in [SYY1] as a map from the complex projective line to the three-dimensional real hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front ([GMM, KUY, KRSUY]), and generic singularities are cuspidal edges and swallowtail singularities. In this paper, we study creations/eliminations of the swallowtails on the image surfaces of the two-parameter family of the confluent hypergeometric differential equations, and give a stratification of the parameter space according to types of singularities. Such a study was made for a 1-parameter family of hypergeometric differential equation in [NSYY]. Contents

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle PT ∗ R 2. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows to interprete the beaks perestroika as the apparent contour of a deformation of the Double Whitney Umbrella singularity

We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable A−B−C −D and G2,F4,E6,7,8 Olshanetsky-Perelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratio of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves. Invited talk given at the Workshop ”Superintegrable systems in classical and quantum mechanics”,