12 problems

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…

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…

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 $\mathcal{L}_2^S\subset\operatorname{Sym}_n(\mathbb{C})$ 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…

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.…

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 $Z_n$ be the variety of principal minors of symmetric $n\times n$ matrices, and let $HD$ denote the hyperdeterminantal module, namely the span of the…