The Barth-Van de Ven-Tyurin-Sato Theorem states that any finite-rank vector bundle on the complex projective ind-space P∞ is isomorphic to a direct sum of line bundles. We establish sufficient conditions on a locally complete linear ind-variety X which ensure that the same result holds on X. We then exhibit natural classes of locally complete linear ind-varieties which satisfy these sufficient conditions. © 2015 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

We study singular Fano threefolds of type $ V_{22}$.

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions.

In this paper a six-valued two-dimensional formal group with ring of coefficients Λ2, lying in ΩU[1/2], is constructed. It is proved that the ring Λ2[1/2] coincides with the image of the ring ΩSU[1/2] in the ring ΩU[1/2]. 

In this work we explicitly calculate the syzygies of the quadratic Veronese embedding ℙ (V) ⊂ ℙ (Sym2V) as representations of the group GL(V ). Resolutions of the sheaves Oℙ (V)(i) are also constructed in the category D(ℙ (Sym2 V)). © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

The Pontryagin-van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group X• is endowed with the topology of uniform convergence on compact subsets of X and in the second, with the topology of uniform convergence on totally bounded subsets of X. The corresponding two classes of groups "reflexive in the sense of Pontryagin-van Kampen" are very wide and are so close to each other that it was unclear until recently whether they coincide or not. A series of counterexamples constructed in this paper shows that these classes do not coincide and also answer several other questions arising in this theory. The results of the paper can be interpreted as evidence that the second approach to the generalization of the Pontryagin duality is more natural.

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies a simple condition.

A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated group.

Let k be an algebraically closed field of characteristic zero and Ga = (k, +) the additive group of k. An algebraic variety X is said to be flexible if the tangent space at every regular point of X is generated by the tangent vectors to orbits of various regular actions of Ga. In 1972, Vinberg and Popov introduced the class of affine S-varieties which are also known as affine horospherical varieties. These are varieties on which a connected algebraic group acts with an open orbit in such a way that the stationary subgroup of each point in the orbit contains a maximal unipotent subgroup of G. In this paper the flexibility of affine horospherical varieties of semisimple groups is proved.

The model of statistical physics on a countable amenable group G is considered. For the natural action of G on the space of configurations S^G, and for any closed invariant set X we prove that there exists pressure which corresponds to a potential with finite norm on X (in the sense of the limit with respect to any Følner sequence of sets in G). The existence of an equilibrium measure is established.