### Book

## Foliations 2012

As an application of our previous results we prove theorems of local and global stability of leaves in sense of Ehresmann and Reeb for conformal foliations of codimention $q>2$. It has been shown that for transversally affine foliations the analogous statements on noncompact closed leaves are not valid. We also remind our rusults about local and global stability of compact leaves of foliations with quasi analytical holonomy pseudogroup admitting an Ehresmann connection and corresponding results of other authors.

As an application of our previous results we prove theorems of local and global stability of leaves in sense of Ehresmann and Reeb for conformal foliations of codimention $q>2$. It has been shown that for transversally affine foliations the analogous statements on noncompact closed leaves are not valid. We also remind our rusults about local and global stability of compact leaves of foliations with quasi analytical holonomy pseudogroup admitting an Ehresmann connection and corresponding results of other authors.

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.

Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations

are developed.

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced foliation (M0, FM0) is formed by the fibers of a locally trivial fibration with the standard fiber L0 over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M0 is formed by the fibers of a locally trivial fibration over a circle or over a line.

The geometry of foliations generated by some differentials of Abelian type is considered. The case where all fibers are closed is studied.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.