26 problems

Let the principal minor map send a complex symmetric matrix to the collection of all its principal minors. Let the group $\operatorname{SL}_2(\mathbb{C})^n\rtimes S_n$ act on the t…

Let … be the likelihood function on positive $4 \times 4$ matrices $P=(p_{ij})_{1 \leq i,j \leq 4}$ of rank at most two. Sturmfels's conjecture. The matrix … is a global maximum of…

Let $Q(n,1/2)$ be the adjacency matrix of an Erdős–Rényi random graph with edge probability $1/2$, and let $p_n^{\mathrm{sym}}$ be the probability that $Q(n,1/2)$ is singular. Symm…

Let $Q_n$ be the random symmetric matrix whose upper-diagonal entries are independent Bernoulli random variables. Symmetric determinant conjecture. Almost surely, … This is the sym…

Let $R$ be a ring in which $2$ is invertible, and let $\varphi$ be a symmetric $n\times n$ matrix of rank $r$. For each $i\leq r$, let $I_i$ be the ideal generated by the…

Conjectural characterization. For every $m$ with $3\leqslant m\leqslant n$, at least one of these two inequalities holds. The inequalities express a concavity-type regularity in th…

An alternating sign matrix (ASM) is a square matrix with entries in $\{-1,0,1\}$, with nonzero entries alternating in sign along each row and column, and with every row and column…

Let $E$ be the ground set of matrix entries, and let $\mathcal{B}\subset E$ be a set of size $2n-1$. A symbic tree is a combinatorial object indexing a maximal cone in the tropical…

Let $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ denote the Fano scheme of $k$-planes contained in the space of symmetric $n\times n$ matrices of rank at most $r$. Let $\kappa(s)$ be…

Let $G$ be a simple connected graph on $n$ nodes with $e$ edges, and let $\mathcal{L}_G$ be the associated logarithmically sparse symmetric matrix model. Write…

Let $A=A^T\in\mathbb{R}^{n\times n}$ be symmetric. For an integer $s$ satisfying $1\leq s<d(A)$, let $v_0\in\mathbb{R}^n$ be a unit-norm initial vector with $d(A,v_0)\geq s+1$, and…

Let $\mathcal{L}_2^S\subset\operatorname{Sym}_n(\mathbb{C})$ be the variety of symmetric matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where…

Nonexistence conjecture. There do not exist canonical small resolutions for symmetric rank loci.

Let $V$ be a vector space, let $\boldsymbol{S}^2V$ denote the space of symmetric matrices on $V$, and let $c6(n,a)$ be the maximum likelihood degree associated with symmetric matri…

Let $A=(a_{ij})$ be a positively realizable symmetric matrix, and call a position $(i,j)$ fully supported if, for every $k$ between $i$ and $j$, at least one of $a_{ik}$ and…

Weighted enumeration conjecture. These weighted enumerations satisfy

A real algebraic integer $\lambda$ is representable if there exists an integral symmetric matrix $M$ such that the map $\lambda\mapsto M$ is an isomorphism of algebras…

Work over $\mathbb{C}$. For a totally real algebraic integer $\lambda$ of degree $d>1$, let $E_s(n,\lambda)$ be the maximum multiplicity of $\lambda$ as an eigenvalue of an $n$-ver…

Let $A,B,C \in \operatorname{Sym}_r(\mathbb{R})$ be real symmetric matrices of size $r$, and set … Assume that $f$ is a polynomial of degree at least three. Singularity conjecture.…

Codimension-bound conjecture. A reasonable upper bound on the codimension of $\mathcal{S}$ ensuring that every range-compatible linear map on $\mathcal{S}$ is local should be

Guo's conjecture. If $\sigma$ is symmetrically realisable, then, for the perturbations under consideration, the list

Let $S_n$ be the polytope of symmetric doubly-stochastic matrices, and let $h^*(S_n)$ denote its Ehrhart $h^*$-polynomial. A regular unimodular triangulation of $S_n$ should exist,…

Let $M_n^{\mathrm{sym}}$ be a random symmetric matrix whose upper-diagonal entries are iid Bernoulli variables, and let $p_n^{\mathrm{sym}}$ denote its probability of being singula…

Let $n$ be a positive integer, let $[n]=\{1,\ldots,n\}$, and in the diagonal-parametrized Cayley identities take $I=J=[n]\setminus\{i_0\}$ for a fixed $i_0\in[n]$. The resulting Be…

Let $n$ be a positive integer and let $S$ satisfy $1le S\le n^2-1$. Let $\mathcal{A}_n(S)$ be the set of all $n\times n$ symmetric $0$-1 matrices with exactly $S$ entries equal to…