7 problems

Let $X$ be a smooth, projective variety defined over a number field $k$, let $P\in X(k)$, and let $A$ be an ample line bundle on $X$. Let $\alpha(P,A)$ denote the infimum of approx…

Let $M\in\mathfrak{M}(m,n)$ and let $M_1,M_2\in M$ satisfy $|M_1M_2|=1$. A blowup of a facher set is the construction referred to by the source, and a unit distance is said to occu…

For $d\geqslant4$, let $e_d(n)$ denote the minimum number of ordinary hyperplanes determined by a set of $n$ points in the relevant $d$-dimensional geometric setting. Higher-dimens…

Let $d\geq 2$, let $S\subseteq\mathbb{R}^d$ be a set whose $(d-1)$-st Beer index $\operatorname{b}_{d-1}(S)$ is defined, and assume that every p-component of $S$ is contractible. L…

For a set $X\subseteq\mathbb{R}^d$ and an integer $k\geq 0$, define … where $\binom{X}{k+1}$ is the set of $(k+1)$-element subsets of $X$. Let $\operatorname{b}_k(S)$ be the $k$-th…

Let $(\mathbb{T}{M},g)$ be a spin spacetime that is asymptotically flat or Kaluza–Klein, whose Ricci tensor satisfies the null-convergence condition, and that admits an action of…

Let $G$ be the group of non-degenerate, equivalently bijective, affine mappings of $\mathbb R^d$. Let $f_G^d(N,k)$ denote the corresponding maximum number of maps having at least…