4 problems

Ulam's conjecture. There exists $x\in\mathbb{S}^n$ such that

Regularity conjecture. The adjacency relation $c_k$ on ${\mathbb Z}^n$ with $k\le n$ is regular.

Let $A$ be a unital C-algebra with a free action of a non-trivial compact quantum group $(H,\Delta)$. Let $\delta$ denote the coaction on $A$, and let $A*_{\delta}H$ be the equivar…

Let $X$ be a compact Hausdorff space equipped with a continuous free action of a non-trivial compact Hausdorff group $G$. The join $X*G$ carries the diagonal action of $G$. Classic…