Index theory for quasi-crystals I. Computation of the gap-label group

In this paper, we give a complete solution to the gap labelling conjecture for quasi-crystals. The method adopted relies on the index theory for laminations, and the main tools are the Connes-Skandalis longitudinal K-theory index morphism together with the measured index formula.     

Conformal geometry of foliations

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.      

Actions of the groups $${\mathbb C}$$ and $${\mathbb C^*}$$ on Stein varieties

In this paper we present recent results concerning global aspects of and -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of and on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.      

On the stability of equivariant foliations

We prove an equivariant version of the Reeb-Thurston stability theorem for foliations invariant under an action of a discrete group.

     

On transversely flat conformal foliations with good measures

Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely -foliations.

     

Exact solutions of the system of the equations of thermal motion of gas in the rarefied space

We study the model describing thermal motion of gas in the rarefied space. This model can be used, in particular, in the study of the state of the medium behind the front of shock wave after very strong blast, in the study of the processes taking place inside of tornado, in the study of the motion of the gas in outer space. For any given initial distribution of the pressure a specific selection of mass Lagrange variables leads to reduction of the system of differential equations describing this motion to the system, for which the number of independent variables is less on the unit. For the obtained system we found all nontrivial conservation laws of the first order. In addition to the classical conservation laws the system has other conservation laws, which generalizes the energy conservation law. We obtained the exact solutions of this system. These solutions describe a variety of different physical processes taking place in the rarefied medium. Using the symmetry properties of the system we got the generating formulas for the receipt of the new solutions using already found earlier solutions of the system.     

Structural Stability of Planar Polynomial Foliations

We discuss natural notions of structural stability of planar polynomial foliations of fixed degree with respect to perturbation within the same restricted set, within the set of all polynomial vector fields of the same degree, and within the set of smooth vector fields. Characterization theorems for structural stability in the latter two settings are obtained as immediate corollaries of known results. We provide sufficient conditions and separate necessary conditions for structural stability of planar polynomial foliations with respect to perturbation within the set of planar polynomial foliations of the same degree.     

Aspects of the Qualitative Theory of Suspended Foliations

We proved that the set of suspended foliations of a compact manifold M is open in the space of foliations with C r -topology. Relation between sets of suspended foliations with Hausdorff and non-Hausdorff graphs are investigated from topological and set-theoretical points of view. Continuum set of pairwise non-isomorphic C X -diffeomorphisms of 1-dimensional manifold with special properties is constructed and used essentially.     

Infinitesimal orbit type theorem for normalizable actions

Let α:G×M→M$\alpha :G\times M\to M$ be a differentiable action of a Lie group G$G$ on a connected differentiable manifold M$M$. Assume that the local tube theorem holds at every x∈M$x\in M$. Then the following is proved: there is a maximal infinitesimal orbit type, called the principal, such that the union of the orbits belonging to this infinitesimal type is an open and dense set in M$M$.