8 problems

Rapid generation conjecture. There exist absolute constants $C,D$ such that, for every nonabelian finite simple Lie algebra $\operatorname{\mathfrak{g}}$ over $\mathbf{F}_p$ and ev…

Let $M^n$ be a closed Riemannian manifold, let $D(M^n)$ denote its diameter, and let $L(M^n)$ denote the length of its shortest closed geodesic. The universal diameter bound conjec…

Let $\mathfrak{M}(2,n)$ denote planar integral point sets with $n$ points, and call $M$ optimal when it attains the minimum possible diameter among sets in $\mathfrak{M}(2,n)$. No-…

Let $d(2,n)$ be the minimum diameter of a planar integral point set with $n$ points. Consider the two open square-container problems: whether there is a $\mathit{\delta}<1$ such th…

Let $d(m,n)$ denote the minimum diameter of an integral point set in $\mathfrak{M}(m,n)$, and let the approach of the stated main estimate refer to the method used to obtain the pa…

Horizontal-section diameter conjecture. There are absolute constants $C_{1},C_{2}>0$ such that

Let $N$ be a positive integer, and let $\operatorname{Sym}(N)$ and $\operatorname{Alt}(N)$ denote the symmetric and alternating groups of degree $N$, respectively. For a finite gro…

Let $C$ be a pure normal simplicial complex of dimension $d-1$ with $n$ vertices. Normal-complex diameter conjecture. The diameter of $C$ is at most … This conjecture strengthens t…