44 problems

Bean's conjecture. The maximum is attained precisely when $F$, up to multiplication by a complex number, is equivalent under $\operatorname{GL}_2(\mathbb R)$ to $F_{n,n}$. In parti…

Binary-form area lower-bound conjecture. One has

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

Let $\Delta(\varepsilon)$ denote the symmetric subset function studied in the source, with $0\leq\varepsilon\leq1$. Exact formula conjecture for $\Delta$. … for all…

There are $N$ coupon types, with draws made independently according to a probability vector $\mathbf{p}=(p_1,\ldots,p_N)$ in the open simplex … and let $T_N$ be the first time ever…

Center–median conjecture. The center and median of $T_0$ have distance zero:

Let $G$ be a finite connected graph, let $d$ be its diameter, let $W(G)$ denote its Wiener index, and let $C_{2d+1}$ denote the cycle on $2d+1$ vertices. DeLaViña–Waller conjecture…

Finite variance extremality. The equal-probability vector $\nu$ minimizes

Let $n\geq 3$ and consider extremizers of the $p$-Stam inequality among normalized monic real-rooted polynomial pairs $(f,g)$ of degree $n$; write $d_{PQ}$ for their pair mismatch…

Extremal-constant problem. In the previous problem, $c=q_\infty$.

The linear upper-bound conjecture. If $d\geq 1$ and $p\geq 2$, then

Let $\Omega=\mathbb{D}$ be the unit disk, let $\Gamma=\partial\Omega=\mathbb{S}^1$, and let $\mathcal{I}_\alpha^n\subset\mathbb{S}^1$ be the union of $n$ evenly spaced intervals of…

Floor-probability bounds conjecture. For all $n\geq 3$ and all $K\in\operatorname{CCSF}([0,1])$,

Let $n\geq 2$, let $q<p<0$, and let $K\subset\mathbb R^n$ be compact with more than one point. The quantities $\operatorname{Cap}_p(K)$ and $\operatorname{Cap}_q(K)$ are the corres…

Let $1<q<2$ or $2<q\leq\infty$, let $\mathfrak{p}(q)$ be the one-dimensional critical exponent, and let $P_+$ denote the Riesz projection on the unit circle. Extremizer conjecture.…

Fix parameters $p,q$ and a positive integer $M$, and let … For ${\cal N}\in{\cal G}(p,q,M)$, write $f(t;{\cal N})$ for its expected adoption level, and let…

Let $a(n)$ be the maximum of the sum of the squares of the edge-lengths over full grid lattice polygons with $n$ vertices, and let $\alpha(n)$ be the explicit lower-bound function…

Conjecture for $F$ on $[\sqrt{2},2]$. For $x\in[\sqrt{2},2]$,

Let $0<p<1$, and let a solution mean a function solving the extremal problem defined earlier in the paper. Evenness conjecture. Any solution of the extremal problem is even. The cl…

For $0<p<\infty$, consider the extremal problem defining the Paley–Wiener extremal function, and call a function attaining the extremum a solution. Uniqueness conjecture. The extre…

Strut-side perimeter conjecture. The perimeter $L(P)$ satisfies

Let $T$ and $N$ be positive integers, and consider the optimization problem over real-coefficient univalent polynomials in $\mathbb D=\{z:|z|<1\}$ with $T$-fold symmetry and degree…

Vertex attainment conjecture. The maximum

Lieb–Solovej conjecture. If $p>2$, then

Let $G$ be any unweighted directed graph, possibly with loops and bidirected arcs, and let $\alpha_1$ and $\alpha_2$ be two jumping constants. For each jumping constant, write…