Vol 83, No 6 (2019)

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 $X=\mathcal Z_{P^n}$.

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.

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.