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Vol 83, No 6 (2019)
- Year: 2019
- Articles: 5
- URL: https://bakhtiniada.ru/1607-0046/issue/view/7541
Articles
Massey products, toric topology and combinatorics of polytopes
Abstract
In this paper we introduce a direct family of simple polytopes $P^{0} {\subset} P^{1} {\subset}{\kern 1pt}{\cdots}$ such that for any $2 {\leq} k {\leq} n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of theirmoment-angle manifolds$\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $*\subset S^{3}\hookrightarrow…\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}} {\hookrightarrow} {\cdots}$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed.As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$, in the Eilenberg–Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(6):3-62
3-62
Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards
Abstract
The authors have recently introduced the class of topological billiards. Topological billiards are glued fromelementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classicallinearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a singlecanonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within theframework of the Fomenko–Zieschang theory of the classification of integrable systems.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(6):63-103
63-103
Instantaneous blow-up versus local solubility of the Cauchy problem for a two-dimensional equation of a semiconductor with heating
Abstract
We consider the Cauchy problem for a model third-order partial differential equation with non-linearity of the form$|\nabla u|^q$. We prove that for $q\in(1,2]$ the Cauchy problem in $\mathbb{R}^2$ has no local-in-time weaksolution for a large class of initial functions, while for $q>2$ a local weak solution exists.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(6):104-132
104-132
Classification of degenerations and Picard lattices of KählerianK3 surfaces with symplectic automorphism group $D_6$
Abstract
In [1]–[6] we classified the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplecticautomorphism groupsof high order. This classification was not considered for the remaining groups of small order($D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$) because each of these cases requires very long and difficultconsiderations and calculations.Here we consider this classification for the dihedral group $D_6$ of order $6$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(6):133-166
133-166
Smooth solutions of the eikonal equation and the behaviour of local minima of the distance function
Abstract
We study smooth solutions of the eikonal equation. To do this, we investigate the problem ofgeometric-topological properties of the singularities of the distance function and the regular set. Weestablish a connection between the caustic and domains where the number of local minima ofthe distance function is constant. We pose a number of problems about reflecting surfaces bringing light to a singlepoint (a focus) and introduce the notions of generalized ellipsoids and paraboloids.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(6):167-194
167-194