### Working paper

## Algebraic groups whose orbit closures contain only finitely many orbits

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.

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.

Based on selected presentations given at the conference “Morphosyntax of Caucasian Languages” held in December 2006 at the Collège de France (Paris).

The Caucasus is the place with the greatest linguistic variation in Europe. The present volume explores this variation within the tense, aspect, mood, and evidentiality systems in the languages of the North-East Caucasian (or Nakh-Daghestanian) family. The papers of the volume cover the most challenging and typologically interesting features such as aspect and the complicated interaction of aspectual oppositions expressed by stem allomorphy and inflectional paradigms, grammaticalized evidentiality and mirativity, and the semantics of rare verbal categories such as the deliberative (‘May I go?’), the noncurative (‘Let him go, I don’t care’), different types of habituals (gnomic, qualitative, non-generic), and perfective tenses (aorist, perfect, resultative). The book offers an overview of these features in order to gain a broader picture of the verbal semantics covering the whole North-East Caucasian family. At the same time it provides in-depth studies of the most fascinating phenomena.

After an introductory chapter that provides an overview to theoretical issues in tense, aspect, modality and evidentiality, this volume presents a variety of original contributions that are firmly empirically-grounded based on elicited or corpus data, while adopting different theoretical frameworks. Thus, some chapters rely on large diachronic corpora and provide new qualitative insight on the evolution of TAM systems through quantitative methods, while others carry out a collostructional analysis of past-tensed verbs using inferential statistics to explore the lexical grammar of verbs. A common goal is to uncover semantic regularities and variation in the TAM systems of the languages under study by taking a close look at context. Such a fine-grained approach contributes to our understanding of the TAM systems from a typological perspective. The focus on well-known Indo-European languages (e.g. French, German, English, Spanish) and also on less commonly studied languages (e.g. Hungarian, Estonian, Avar, Andi, Tagalog) provides a valuable cross-linguistic perspective.

The article deals with the problem of the author as the subject of consciousness expressed through the text in its entirety. Special emphasis is laid on modality revealed in the author’s evaluation of events, characters and the world in general.

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.