12 problems

Let $M$ and $N$ be smooth closed manifolds of dimensions $p$ and $q$, respectively. Write … A $(\mathbb{Z}/2)^2$-equivariant map is understood with respect to the involutions excha…

Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\geq 4$, and $n$ even, and let $N_0$ denote a punctured version of $N$. For an embedding…

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $(X,T)$ be a topological dynamical system, let $d$ be a positive integer, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topological…

Let $N^n_m$ be the subset of $\mathbb{R}^n$ consisting of points with at most $m$ rational coordinates. Chigogidze's conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ can…

The Prem Conjecture. Smoothly $k$-realizable general-position smooth maps $N^n\to\mathbb{R}^m$ are smooth $k$-prems, at least in the metastable range

Let $s,d,r\ge2$ be integers satisfying … and let $K$ be a finite $s$-dimensional simplicial complex. An almost $r$-embedding is a map $f:K\to\mathbb R^d$ such that…

Let $D=D^{s_1}\sqcup\ldots\sqcup D^{s_r}$ be a disjoint union of $r$ disks, let $f:D\to B^d$ be a proper PL map such that…

Let $K^n$ be a simplicial complex, let $\epsilon$ be positive, and let $B^d_1\subset\mathbb{R}^d$ denote the unit ball. A piecewise smooth embedding of $K^n$ into $\mathbb{R}^d$ ha…

Let $X$ be a $k$-dimensional simplicial complex with $N$ simplices, and suppose that each vertex lies in at most $L$ simplices. Let $n\ge 2k+1$. An embedding is 1-thick if the dist…

Let $X$ be a compact length metric space of finite Hausdorff dimension. A map from $X$ into a Euclidean space has bounded length distortion if it distorts lengths by a controlled r…