54 problems

Let $w=[x,y]=x^{-1}y^{-1}xy$ be the commutator word, and for a group $G$ let $w(G)$ be its verbal set. The word $w$ has width $N$ in $G$ if the verbal subgroup generated by $w(G)$…

Böttcher–Wenzel conjecture. The commutator satisfies

Let $G$ be a finite simple group. Coprime-order commutator conjecture. Every element of $G$ is a commutator of two elements of coprime orders. This conjecture would extend the Ore…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conje…

Let $\mathbb{K}=\mathbb{R}$ and let $B_1,\ldots,B_m\in M(n,\mathbb{K})$, where $M(n,\mathbb{K})$ is the space of $n\times n$ matrices over $\mathbb{K}$. Let $[A,B]=AB-BA$ and let…

Etingof–Kim–Ma conjecture.

Let $c_n$ be the $n$th Zassenhaus exponent, written as a sum over words $W(t_1,\ldots,t_n)$ in the letters $a$ and $b$, with $c(W)$ denoting the coefficient of the word and…

Let $T$ satisfy the same conditions as in Theorem 1.2 of the source, let $U$ and $V$ be matrix $A_p$ weights, let $b$ be a scalar function, and let $m\in\mathbb{N}$. Write…

Let $X$ be a Banach space. A Wintner space is a Banach space for which every non-commutator in $\mathcal{B}(X)$ has the form $\lambda I+K$, where $lambda e 0$ and $K$ belongs to a…

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a $p$-element. Suppose that for every $g\in G$, the commutator $[x,g]$ is either $1$ or $p$-singular, meaning tha…

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ have order $p$. Suppose that for every $g\in G$, the commutator $[x,g]$ is either $1$ or $p$-singular, meaning that…

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose and let $[A,B]=AB-BA$. Assume … for…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Ge–Li–Lu–Zhou conjecture.…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Ge–Li–Lu–Zhou conjecture.…

Let $\mathbb{K}=\mathbb{R}$ and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose, let $\bot$ denote orthogonality for the Frobenius inner produc…

Commutator Baer–Suzuki conjecture. If $[x,g]$ is either $1$ or $p$-singular for every element $g\in G$, then $x\in O_p(G)$.

Multicommutator Baer–Suzuki conjecture. If, for some integer $k\geqslant 1$, $ab$ is a $p$-element for all $a,b\in\Gamma_k(x)$, then $x\in O_p(G)$.

Let $\operatorname{CL}_{\lVert\!\lVert\!\lVert\cdot\rVert\!\rVert\!\rVert}([0,\infty))$ denote the class of commutator-Lipschitz functions for the chosen unitarily invariant norm,…

Let $\widehat{\mathcal M}_+$ be the space of Fourier transforms of finite positive Borel measures on $\mathbb R$, and define … Let $f$ be a non-negative function that is operator m…

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ and that $X$ is compact with finite unitarily invariant norm…

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ are compact with finite unitarily invariant norm…

Hecke deformation conjecture. Corollary 28 and Corollary 29 both seem to hold in $\mathcal{H}$ when every $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

Optimal exponent conjecture. The smallest $m\in\mathbb{N}$ such that

Let $P$ and $Q$ be the momentum and position operators, and let $f$ and $g$ be bounded, real, measurable functions. Define … For a positive parameter $\alpha$, let $K_{\alpha}$ den…

Let $1<q<p<\infty$, let $\mu\in A_p$ and $\lambda\in A_q$, and let $T$ be a Calderón–Zygmund operator. Consider the commutator $[b,T]$ acting from $L^p(\mu)$ to $L^q(\lambda)$, and…