28 problems

Representability conjecture for coordinate q-matroids. If the corresponding matroid of a coordinate $q$-matroid is representable, then the $q$-matroid is representable.

Let $q$ be a prime power. A matroid is $GF(q)$-representable if it has a representation over the finite field $GF(q)$, and two such representations are considered equivalent when t…

Finite-dimensional representability conjecture. There exists a finite-dimensional $W$-(super)algebra $B$ such that

Pure-state density representability conjecture. One may naturally conjecture that

Nation–Picketing's conjecture. Having a distributive skeleton should be sufficient for $L$ to be type-1 representable.

A finite relation algebra is a finite algebra equipped with the Boolean operations, relational composition, converse, and identity element satisfying the relation-algebra axioms; i…

Let $C_3$ denote the universal constant multiplying the term involving the deformation of the velocity field in the three-dimensional density-current representability bound. Optima…

Representability conjecture for K(r,t). Every matroid obtained from $K(r,t)$ by relaxing a circuit-hyperplane into a basis is representable.

Walsh's lift conjecture. Every lift of $M$ is isomorphic to $M^N$ for some matroid $N$ on the circuits of $M$.

Let $d$ be a nonnegative integer, and let $\mathsf{K}$ be a simplicial complex. A simplicial complex is $d$-Matoušek and $d$-Leray as defined in the source paper. The d-Matoušek-to…

Let $\mathcal{P}_{M,N}$ denote the space of densities for an $M$-vertex graph and $N$ particles, and call a density universally $v$-representable (uv) if it arises from the relevan…

Let $\bm{w}$ be a weight vector with distinct entries, and let $\mathsf{\Pi}(\bm{w},N,d)$ denote the polytope of spectra associated with the weighted $1$-body $N$-representability…

Affine-plane representability problem. Determine whether

Representability bound conjecture. If $h(X)\leq d$ for $d\geq 1$, then

Let $M$ be a matroid. A matroid is representable over all fields of size at least $4$ when it has a representation over every field whose cardinality is at least $4$. Let…

For a prime power $q$, let $\mathcal{AC}_q$, $\mathcal{SL}_q$, and $\mathcal{AF}_q$ be the corresponding classes of matroids, with extremal functions $h_{\mathcal{AC}_q}(r)$,…

Let $\mathcal{GM}$ denote the class of golden-mean matroids, and let $h_{\mathcal{GM}}(r)$ be its extremal function, giving the maximum size of a rank-$r$ matroid in $\mathcal{GM}$…

Let $M_1$ be a matroid with a circuit-hyperplane $H$, and let $M_2$ be obtained from $M_1$ by relaxing $H$. If both $M_1$ and $M_2$ are representable over a finite field…

Finite-minimality conjecture. There are only finitely many $2$-polymatroids that are minimal with the property of being non-representable over $\mathbb F$ under these reduction ope…

Let $S\subseteq\mathbb{R}^n$ be MICP representable, so that … where $C\subseteq\mathbb{R}^d$ and each $A_{\bm{z}}\subseteq\mathbb{R}^d$ is a projection of a closed convex set. A fi…

For a fixed finite field $\mathbb{F}$, the fraction of matroids on an $n$-element ground set that are representable over $\mathbb{F}$ tends to zero as $n\to\infty$. The folklore co…

Nonrepresentability conjecture. Asymptotically almost every matroid is not representable over any field.

Missing axiom conjecture. There is no characterisation of the class of representable matroids, and no characterisation of the class of matroids representable over an infinite field…

A matroid is representable over a field when it arises from a finite set of vectors over that field, with independent sets corresponding to linearly independent vector subsets. A m…

Structural conjecture. Either $M$ is ternary, for some $e\in L$ the deletion $M\backslash e$ is binary, or $M$ has an $F_7$-minor and either an $F_7^-$- or $(F_7^-)^*$-minor.