20 problems

Flexibility-free approximate polymorphism conjecture. The Alekseev--Filmus theorem should hold without the assumption that $P$ is flexible, for any alphabet $\Sigma$: for every suf…

Zonotope sparsification conjecture. There exists a matrix $W\in\mathbb{R}^{d\times n}$ such that

Let $F$ and $H$ be graphs. Let an $F$-forest $T$ be the forest construction used in the source, and let $T^{\star}$ denote its associated graph. Forest sufficient-condition conject…

Let $F$ and $H$ be graphs. An $F$-forest is the forest-like construction used in the source, and $T^{\star}$ denotes its associated graph. Forest-characterisation conjecture. The c…

Let $F$ and $H$ be graphs, and let $M_{F,H}(\varepsilon)$ denote the relevant minimum target-size function for approximate homomorphisms to $F$-homomorphism-free graphs. Let…

Barto–Batistelli–Berg's conjecture. Finding an LO $\ell$-colouring of a 3-uniform hypergraph that admits an LO $k$-colouring is $\operatorname{NP}$-hard for every constant…

A graph is $c$-colourable if its vertices can be assigned one of $c$ colours so that adjacent vertices receive different colours. For fixed constants $k\geq c\geq 3$, consider the…

Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$, and let $\epsilon>0$. A mod-$p$ condition on $S^n$ is one of the conditions from the original family under cons…

The approximate graph colouring problem asks, for fixed integers $3\leq c\leq d$, whether a given graph is $c$-colourable and, if so, to find a $d$-colouring. In its decision varia…

Let $K\subseteq\mathbb{R}^d$ be a zonotope and let $0<\varepsilon\leq\frac12$. A zonotope is a Minkowski sum of finitely many line segments; its number of segments is the number of…

Let $\gamma$ be a polynomially convex simple closed curve in $\mathbb{C}^N$. A simple closed curve is locally polynomially convex if each of its points has a neighborhood whose int…

EP approximates each site $f_i(x)$ by a Gaussian site-approximation $q_i(x\mid r_i,\beta_i)$, iteratively updating the sites until reaching a fixed point; the resulting Gaussian ap…

Approximate complete-class simplification conjecture. Further simplifications can be obtained by allowing approximate versions of complete class theorems, Bayesian estimators, opti…

Let $X$ be a quasi-compact and quasi-separated algebraic stack. An approximation of $X$ is a factorization … where $X\to X_0$ is affine and $X_0$ is of finite presentation over…

Let $X$ be a quasi-compact and quasi-separated algebraic stack. A quasi-coherent $\mathcal{O}_X$-module is said to have the completeness property when it is a directed colimit of f…

For $n$ voters, the normalized Banzhaf power vectors of weighted majority games are points in the unit simplex. The interior of the unit simplex consists of normalized nonnegative…

Let $n\ge3$, let $\sigma\in\mathbb{R}_{\ge0}^n$ satisfy $\Vert\sigma\Vert_1=1$, and write $\sigma^n$ for the desired power distribution in dimension $n$. Shapley–Shubik approximati…

Let $[q;w_1,\dots,w_n]$ be a weighted majority game with $q\in(0,1)$ and $w\in\mathbb{R}_{\ge0}^n$ normalized by $\Vert w\Vert_1=1$. Let $\Delta=\max_{1\le i\le n}w_i$, and let…

Let $P$ be a power index, let $[q;w_1,\dots,w_n]$ denote a weighted majority game with $q\in(0,1)$, normalized nonnegative weights $w\in\mathbb{R}_{\ge0}^n$ satisfying…

Let $X$ be a finite connected $CW$-complex and let $\overline{X} \to X$ be a $G$-covering. Approximation conjecture for $L^2$-torsion. If the $G$-$CW$-structure on $\overline{X}$ a…