91 problems

Grone–Merris conjecture. The eigenvalue sequence of $\Delta_0$ is majorized by the conjugate degree sequence:

Duval–Reiner conjecture. For every $q$-uniform family $F$ and every $r\ge1$,

Let $n\geq2$ and let $\mu,\lambda\in\mathbb{N}^{m}$ satisfy $|\mu|=|\lambda|=d$. For the term-normalized complete homogeneous symmetric functions $H_{n,\mu}$ and $H_{n,\lambda}$, w…

Nielsen's incomparability conjecture. The relative volume of pairs of LOCC-convertible bipartite pure states vanishes in the limit of large dimensions; equivalently, for two indepe…

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm dim}} \mathcal X={{\rm dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector…

Let $A$ be a Hermitian matrix, and let $\mathcal{X}$ and $\mathcal{Y}$ be subspaces of the same dimension, with orthonormal basis matrices $X$ and $Y$, respectively. Let…

Let $z_i$, $1\le i\le n$, be distinct points in $\mathbb{C}$ and let $a_i>0$, $1\le i\le n$, satisfy $\sum_{i=1}^{n}a_i=1$, where $n\ge 2$. Let $w_j$, $1\le j\le n-1$, be the zeros…

Grone–Merris conjecture for relative pairs. The relative Laplacian spectrum is majorized by the conjugate relative degree sequence:

Majorization conjecture. If $T\in\mathcal{A}'$ and $n\ge 1$, then for every $P\in\mathcal{C}_n$ at least one of the relations

Let $(\mathcal K,\mathcal K')$ be a family pair, with relative Laplacian spectrum $\mathbf{s}(\mathcal K,\mathcal K')$ and transposed relative generalized degree sequence…

Let $d4a6$ be the $2$-family of edges of an arbitrary graph. Write its Laplacian spectrum as the weakly decreasing sequence $d4a2(d4a6)=(s_1,s_2,d4d5)$ and let…

de Bruijn–Springer conjecture. The inequality

Complex spectral-order conjecture. If $P\in\mathcal{C}_{n}$, then

Noncommutative alpha-geometry conjecture. Suppose $n\geq2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then…

Commutative alpha-geometry conjecture. Suppose $n>2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then $\operatorname{Scal}_p$…

Let $K$ be an $r$-dimensional simplicial complex. Using the spectrum $\bm{\lambda}_{r-1}(K)$ of the up-Laplacian and the conjugate vertex-degree sequence $mathbf d0^top(K)$, with…

Let $\mathcal{M}$ be a semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$, and let $\mathcal{P}$ be the positive cone of the space of all $\tau$-…

Let $G$ be a graph with Laplacian eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$, and let $d_k'$ be the number of vertices of $G$ having degree at least $k$. The sequence…

Let $p=(p_1,\ldots,p_N)$ be the popularity vector of an independent-request move-to-front or LRU model, and let $C$ be a cache capacity. The search cost is the position of the requ…

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

Let $A,B\in\mathbb{H}_n$ be Hermitian matrices, and let $H$ and $R_k$ denote the operators defined in the paper from the coefficients of $e^{(A+B)t}$ and $e^{At}e^{Bt}$, respective…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. Let $P_\lambda(x;q,t)$ be the Macdonald polynomial, $\mathcal M_S(\mathbb F_{\geqslant0})$ the Muirhead semiring over…

Let $\lambda$ and $\mu$ be partitions, and let $\mathcal J_C^\sigma(\mathbb R_{\geqslant0})$ be the Jack cone generated by normalized Jack differences with parameter $\sigma$. Fix…

Fix $\tau_0\in[0,\infty]$, and let $\lambda$ and $\mu$ be partitions. Let $\mathcal M_S(\mathbb R_{\geqslant0})$ denote the Muirhead semiring over the nonnegative reals. Muirhead-s…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. For Jack polynomials $P_\lambda(x;\tau)$, let $\mathbb F_{\geqslant0}^{\mathbb R}$ denote the real cone of rational fu…