### Working paper

## Infinite transitivity and special automorphisms

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we recover the description of the image of the moment map of $T^*X$ obtained by Knop. In our proofs we use only geometric methods which do not involve differential operators.

In the paper I consider the causative constructions in Russian. I examine the use of tense and aspect in constructions with the verbs *zastavit’ */ *zastavljat’ *‘make’ and *pozvolit’ */ *pozvoljat’ *‘let, allow’. I also include the verb *delat’ */ *sdelat *‘make’ in my analysis, though this verb has special syntactic and semantic characteristics.

The striking feature of the causative constructions with eventive subjects is that the tensed forms and temporal adverbs in these constructions do not obligatorily refer to the causing situation. The tensed forms and adverbials sometimes refer only to the caused situation.

I assume that it is the nature of events vs. participants that is responsible for these distinctions. Each dynamic event is associated with some result. I have shown that in some cases what the tense of the causative verb and temporal adverbials refer to is the result of the causing event, and not the causing event in the narrow sense.

In my article, I address the factors which favor using a verb as labile (both transitive and intransitive, with no formal change required).

Haspelmath (1993) proposes that the key feature which conditions a way of marking (in)transitivity of verbs in the transitive / intransitive verb pair is the spontaneity parameter.

However, the statistical analysis of Haspelmath’s data shows that for labile / ambitransitive verb the main parameter is the lexical semantic class of the verb, not the degree of spontaneity. This lets us discover a more general principle: phenomena which are not purely grammatical, but rather lexico-grammatical (as lability) depend on lexical features, not on generalized grammatical or semantic parameters.

Fascinating and surprising developments are taking place in the classification of algebraic varieties. Work of Hacon and McKernan and many others is causing a wave of breakthroughs in the Minimal Model Program: we now know that for a smooth projective variety the canonical ring is finitely generated. These new results and methods are reshaping the field. Inspired by this exciting progress, the editors organized a meeting at Schiermonnikoog and invited leading experts to write papers about the recent developments. The result is the present volume, a lively testimony of the sudden advances that originate from these new ideas. This volume will be of interest to a wide range of pure mathematicians, but will appeal especially to algebraic and analytic geometers.

The paper looks into the contemporary state of the problem of decision-making and preference of some alternatives over others, discussing intransitivity of relations of superiority: one object is superior to another in a certain aspect, while the second is superior to the third and the third is superior to the first (A>B, B>C, C>A). The authors analyze two broad groups of theories and empirical studies reflecting opposite views on the nature of the relations and rationality of intransitivity of relations of superiority. The authors argue that understanding of intransitivity of superiority relations is no less important line of cognitive development than understanding of transitivity; they should be studied as complementary subjects. Thus it is necessary to study individual differences in cognitive sets with regard to transitivity/intransitivity of superiority, as well as individual characteristics of solving problems of that kind.

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.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

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.