11 problems

Lutz–Nevo conjecture. The following are equivalent:

A tame quasiconformal manifold is a topological manifold equipped with a quasiconformal structure whose chart transition functions are tame quasiconformal mappings. Tame quasiconfo…

The piecewise linear Poincaré conjecture. Every piecewise linear homotopy $4$-sphere is piecewise linear homeomorphic to $S^4$.

Let $C$ be a $c$-chain in $\mathbb{R}^d$ with $c<d$, and let $U$ be a finite set of points in $\mathbb{R}^d$ in strong general position with respect to $C$. Simplicial-chain realiz…

Let $c,n,d$ be non-negative integers such that $c<d$. Let $C$ be a $c$-chain in $d53d^d$, let $N$ be an $n$-dimensional closed PL manifold, and let $f:N\tod53d^d-\partial C$ be a P…

PL-regularity conjecture. For any reasonable class of finite PL functions, a $0$-cell is far more likely to be PL regular than PL critical.

Generic total H-complexity conjecture. With probability $1$, the total $H$-complexity of $F$ is less than or equal to the number of $0$-cells of $\mathcal{C}(F)$.

Coarse bounded H-complexity conjecture. The coarse bounded sublevel $H$-complexity (respectively, the coarse bounded superlevel $H$-complexity) of $F$ is a lower bound for the glob…

A topological space may admit distinct notions of equivalence: topological isomorphism means homeomorphism, while combinatorial isomorphism means piecewise linear homeomorphism. Th…

PL analogue conjecture. The piecewise linear (PL) analogue of Theorem 1 holds.

Let $(K3)_{16}$ and $(K3)_{17}$ denote the simplicial complexes with $16$ and $17$ vertices, respectively, constructed in the dilatation process described in the paper. The PL-equi…