In the theory of affine SL(2)-embeddings, which was constructed in 1973 by Popov, a locally transitive action of the group SL(2) on a normal affine three-dimensional variety X is determined by a pair (p/q, r), where 0 < p/q ≤ 1 is a rational number written as an irreducible fraction and called the height of the action, while r is a positive integer that is the order of the stabilizer of a generic point. In the present paper it is shown that the variety X is toric, that is, it admits a locally transitive action of an algebraic torus if and only if the number r is divisible by q - p. For that, the following criterion for an affine G/H-embedding to be toric is proved. Let X be a normal affine variety, G a simply connected semisimple group acting regularly on X, and H ⊂ G a closed subgroup such that the character group X(H) of the group H is finite. If an open equivariant embedding G/H → X is defined, then X is toric if and only if there exist a quasitorus T̂ and a (G × T̂)-module V such that X ≅G V//T̂. In the substantiation of this result a key role is played by Cox's construction in toric geometry.

Studying the dynamics of a flow on surfaces by partitioning the phase space into cells with the same limit behaviour of trajectories within a cell goes back to the classical papers of Andronov, Pontryagin, Leontovich and Maier. The types of cells (the number of which is finite) and how the cells adjoin one another completely determine the topological equivalence class of a flow with finitely many special trajectories. If one trajectory is chosen in every cell of a rough flow without periodic orbits, then the cells are partitioned into so-called triangular regions of the same type. A combinatorial description of such a partition gives rise to the three-colour Oshemkov-Sharko graph, the vertices of which correspond to the triangular regions, and the edges to separatrices connecting them. Oshemkov and Sharko proved that such flows are topologically equivalent if and only if the three-colour graphs of the flows are isomorphic, and described an algorithm of distinguishing three-colour graphs. But their algorithm is not efficient with respect to graph theory. In the present paper, we describe the dynamics of Ω-stable flows without periodic trajectories on surfaces in the language of four-colour graphs, present an efficient algorithm for distinguishing such graphs, and develop a realization of a flow from some abstract graph.

We prove that the bound from the theorem on 'economic' maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound \(dn + n + 1)/(m - n - d)] + d from the theorem on 'economic' maps. Bibliography: 16 titles.

We study the limit as α → 0+ of the long-time dynamics for various approximate α-models of a viscous incompressible fluid and their connection with the trajectory attractor of the exact 3D Navier-Stokes system. The α-models under consideration are divided into two classes depending on the orthogonality properties of the nonlinear terms of the equations generating every particular α-model. We show that the attractors of α-models of class I have stronger properties of attraction for their trajectories than the attractors of α-models of class II. We prove that for both classes the bounded families of trajectories of the α-models considered here converge in the corresponding weak topology to the trajectory attractor 0 of the exact 3D Navier-Stokes system as time t tends to infinity. Furthermore, we establish that the trajectory attractor α of every α-model converges in the same topology to the attractor 0 as α → 0+. We construct the minimal limits min ⊆ 0 of the trajectory attractors α for all α-models as α → 0+. We prove that every such set min is a compact connected component of the trajectory attractor 0, and all the min are strictly invariant under the action of the translation semigroup. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We present a new justification logic corresponding to the Gödel-Löb provability logic GL and prove the realization theorem connecting these two systems in such a way that all the realizations provided in the theorem are normal.

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid with respect to an index 2 subgroup of its automorphism group. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N->S^m. We study the specific case of knotted tori, i. e. the embeddings S^p x S^q -> S^m. The classification of knotted tori up to isotopy in the metastable dimension range m>p+3q/2+3/2, p<q+1, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension:

Theorem. Assume that p+4q/3+2<m<p+3q/2+2 and m>2p+q+2. Then the set of smooth embeddings S^p x S^q -> S^m up to isotopy is infinite if and only if either q+1 or p+q+1 is divisible by 4.

Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new beta-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

We analyze compact multiplicative semigroups of affine operators acting in a finite-dimensional space. The main result states that every such semigroup is either contracting, that is, contains elements of arbitrarily small operator norm, or all its operators share a common invariant affine subspace on which this semigroup is contracting. The proof uses functional difference equations with contraction of the argument. We look at applications to self-affine partitions of convex sets, the investigation of finite affine semigroups and the proof of a criterion of primitivity for nonnegative matrix families.

Continuous Morse-Smale flows on closed manifolds whose nonwandering set consists of three equilibrium positions are considered. Necessary and sufficient conditions for topological equivalence of such flows are obtained and the topological structure of the underlying manifolds is described. Bibliography: 36 titles. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We study the Cox realization of an affine variety, that is, a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results in the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of an affine toric variety of dimension ≥ 2 without nonconstant invertible regular functions has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to the Anick automorphism of the polynomial algebra in four variables.

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.

We classify $\mathbb Q$-Fano threefolds of Fano index > 2 and sufficiently big degree.

We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.

We prove that every complete foliation (M, F) of codimension q > 1 is either Riemannian or a (Conf (S^q), S^q)-foliation. We further prove that if (M, F) is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In particular, every proper conformal non-Riemannian foliation (M, F) has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all non-closed leaves is a connected q-dimensional orbifold. We show that every countable group of conformal transformations of the sphere S^q can be realized as a global holonomy group of complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly infinite-dimensional) representations without any topological structure. In addition, we prove that for certain induced representations, irreducibility is implied by Schur irreducibility. Both results are obtained in a more general form for representations over an arbitrary field © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

It is demonstrated that every (0, 1)-matrix of size n×m having Boolean rank n contains a column with at least √n/2 − 1 zero entries. This bound is shown to be asymptotically optimal. As a corollary, it is established that the size of a full-rank Boolean matrix is bounded from above by a function of its tropical and determinantal ranks.

We consider the Paley--Wiener spaces of L2-functions whose Fourier transform has a bounded support. We show that every continuous mapping that generates a superposition operator acting on these spaces is affine and injective.