A. Skopenkov

A. Skopenkov

Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a skew-symmetric $n\times n$-matrix $A$ with integer entries, whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$, $\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and $\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $σ,τ$ of $K$ the entry $A_{σ,τ}$ equals to the algebraic intersection of $fσ$ and $fτ$. We prove some analogues of this result (for any parity of $k$), including those for $\mathbb Z_2$- and $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds. The main novelty of this paper is passing from the cohomology condition of Paták-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl).

S. Dzhenzher, T. Garaev, O. Nikitenko et al.

The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).

S. Dzhenzher, A. Skopenkov

We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $\varphi_1,\ldots,\varphi_5 : [\,0, 1\,]\to [\,0,1\,]$ such that for any continuous function $f: [\,0,1\,]^2\to\mathbb R$ there is a continuous function $h: [\,0,3\,]\to\mathbb R$ such that for any $x,y\in [\,0, 1\,]$ we have $$f(x,y)=\sum\limits_{k=1}^5 h\left(\varphi_k(x)+\sqrt{2}\,\varphi_k(y)\right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.