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Of all publications in the section: 120
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Article
Айзенберг А. А. Математический сборник. 2017. Т. 208. № 9. С. 3-25.

Manifolds with locally standard half-dimensional torus actions represent a large and important class of spaces. Cohomology rings of such manifolds are known in particular cases but in general even Betti numbers are difficult to compute. Our approach to this problem is the following: we consider the orbit type filtration on a manifold with locally standard action and study the induced spectral sequence in homology. It collapses at a second page only in the case when the orbit space is homologically trivial. The cohomology ring in this case was already computed. Nevertheless, we can completely describe the spectral sequence under more general assumptions, namely when all proper faces of the orbit space are acyclic. The theory of sheaves and cosheaves on finite partially ordered sets is used in the computation. The second page of the spectral sequence can be described as the cohomology of a certain sheaf on the dual simplicial poset, whose value on a simplex is the homology of the corresponding toric orbit. We study this and related sheaves and establish the generalizations of the Poincare duality and the Zeeman-McCrory spectral sequence for sheaves of ideals of exterior algebras.

Article
Васильев В. А. Математический сборник. 2016. Т. 207. № 10. С. 4-27.

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions of strictly hyperbolic PDE's with constant coefficients in R^N) at the parabolic singular points of their wavefronts (i.e., at the points of types P_8^1, P_8^2, +X_9, -X_9, X_9^1, X_9^2, J_10^1, J_10^3). These points form the next difficult class of the natural classification of singular points after the so-called simple singularities A_k, D_k, E_6, E_7, E_8, studied previously. Also we promote a computer program counting for topologically different Morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.

Article
Авербух Ю. В. Математический сборник. 2015. Т. 206. № 7. С. 3-32.
Article
Рыбаков С. Ю., Трепалин А. С. Математический сборник. 2017. Т. 208. № 9. С. 148-170.
Article
Горчинский С. О., Осипов Д. В. Математический сборник. 2015. Т. 206. № 9. С. 21-98.
Article
Айзенберг А. А., Бухштабер В. М. Математический сборник. 2021.

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{\St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove this space to be a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{\St_n,\lambda}/T^n$ is not a polytope, hence $M_{\St_n,\lambda}$ is not a quasitoric manifold. However, there is a natural permutation action on $M_{\St_n,\lambda}$ which induces the combined action of a semidirect product $T^n\rtimes\Sigma_n$. The orbit space of this large action is a simple polytope $\B^n$. The structure of this polytope is described in the paper. In case $n=3$, the space $M_{\St_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{\St_3,\lambda}$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{\St_3,\lambda}$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.

Article
Аржанцев И. В., Куюмжиян К. Г., Зайденберг М. Г. Математический сборник. 2012. Т. 203. № 7. С. 3-30.

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.

Article
Жужома Е. В., Медведев В. С. Математический сборник. 2016. Т. 207. № 5. С. 69-92.
Article
Белошапка И., Горчинский С. О. Математический сборник. 2016. Т. 207. № 1. С. 45-72.
Article
Гущин А. А. Математический сборник. 1982. Т. 118. № 2. С. 164-172.
Article
Вьюгин И. В., Шкредов И. Математический сборник. 2012. Т. 203. № 6. С. 81-100.
Article
Богатая С. И., Богатый С. А., Кудрявцева Е. А. Математический сборник. 2012. Т. 203. № 4. С. 103-118.

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.