114 problems

For each $k$, let $\mathsf{M}_k^{{||}}$ and $\mathsf{M}_k^{{\ddagger}}$ denote the parallel and series transition matrices, and let $\mathrm{Eul}_{k}(t)$ be the Eulerian polynomial…

For each $k$, let $N_k$ be the matrix appearing in the expression for the generating series $\mathcal{R}_k(t)$, and write $\pi_{N_k}(t)$ and $\chi_{N_k}(t)$ for its minimal and cha…

Let $q$ be a prime power, let $\sigma$ be an operator with $\sigma^n=\operatorname{id}$, and let $W_1$ and $W_2$ be $\sigma$-invariant subspaces satisfying … Write $h_{\sigma}(V)$ for…

Let $C\in\mathbb{C}^{d\times d}$ be a scaling matrix of rank four. A principal $4$-minor is non-vanishing when the corresponding principal submatrix has nonzero determinant. Rank-f…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $S(2n,\mathbb{R})$ denote the space of real symmetric $2n\times 2n$ matrices, let $J$ be the standard symplectic matrix, and let $\rho$ be the symplectic-group action on…

Let $p$ and $q$ be monic polynomials of degree $2$, let $V$ be a vector space, and let $(b,u)$ be a symplectic pair on $V$. A symplectic $(p,q)$-difference is a symplectic pair sat…

Let $U$ and $V$ be finite-dimensional vector spaces over a field $\mathbb{F}$. Let $\mathcal{S}$ be a linear subspace of $\mathrm{Hom}(U,V)$. We say that it is intransitive when…

Alternating-form symmetric-operator equality conjecture. There exists a maximal partially complete $b$-singular flag $\mathcal{F}$ of $V$ such that

Let $V=\bigoplus_i V_i$ be a finite-dimensional vector space, with each $V_i$ equipped with a non-degenerate symmetric bilinear pairing, and let $K\subset V$ be a subspace such tha…

For every integer $n\ge 1$ and every convex body $K\subset\mathbb{R}^n$, let $E$ be a maximum-volume ellipsoid contained in $K$, let $c$ be the center of $E$, and define…

For every unitarily invariant matrix norm $\|\cdot\|_{\mathrm{ui}}$, does there exist a constant $c>0$, independent of $n$ and of $G\in\mathbb{C}^{n\times n}$, such that … Here…

For a matrix $M\in\{0,1\}^{n\times m}$, define its binary rank by…

For every field $\mathbb{F}$, every integer $n\geq 1$, and every family $S\subseteq M_n(\mathbb{F})$, if there exists an integer $\ell\geq 1$ such that…

For a graph $G$, let $\mathcal{S}(G)$ be the set of real symmetric matrices whose off-diagonal zero–nonzero pattern is the adjacency pattern of $G$. Define $MB(G)$ to be the minimu…

For every integer $n\ge 1$ and every matrix $A\in E_0^f\cap Q_0$ of order $n$, all principal minors of $A$ are nonnegative; equivalently, $A\in P_0$, that is, $\det A[I,I]\ge 0$ fo…

Classify, up to standard equivalence, all matrices $H\in\mathbb{C}^{6\times 6}$ satisfying $|H_{ij}|=1$ for every $i,j$ and $HH^{*}=6I_{6}$. The conjecture asserts that every such…

Chowla matching conjecture. If $B$ is a Chowla subspace, then $A$ is matched to $B$.

Let $X$ and $Y$ be two $n\times n$ matrices, let $[X,Y]=XY-YX$ be their commutator, and let $||\cdot||$ denote the Hilbert–Schmidt norm. Böttcher–Wenzel conjecture. … The conjectur…

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Let $A_1,B_1,C_1,A_2,B_2,C_2$ be finite-dimensional vector spaces over a field $\mathbb{K}$, and let $T_1\in A_1\otimes B_1\otimes C_1$ and $T_2\in A_2\otimes B_2\otimes C_2$. Thei…

Continuous orthonormal trivialization conjecture. The bases can be chosen so that the map $T:G\to GL_d(\mathbb{K})$ defined by

Tasaka's isomorphism conjecture. The map $\widetilde{\xi}$ is an isomorphism. The conjecture concerns the depth-three case and is attributed to Tasaka. The paper does not report a…