46 problems

Let $f$ be a $C^r$ diffeomorphism and let $\mathcal Z^r(f)$ denote its $C^r$-centralizer. Smale's conjecture. For $C^r$-generic $f$, the centralizer is trivial, namely … The source…

Let $X$ be a compact homogeneous manifold and let $f_0$ be an affine $K$-system on $X$. Assume that $\mathcal{Z}^\infty(f_0)$ contains an embedded subgroup isomorphic to…

Let $M$ be a compact connected manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $f:M\to\mathbb{R}$ be a function that is either $C^1$ or $C^0$. Call $f$ invariant…

Let $A$ be a hyperbolic toral automorphism and let $f$ be a volume-preserving, possibly non-ergodic, $C^1$-small perturbation of $T_A\times\operatorname{id}$. Write…

Let $m,n\geq 1$ be fixed integers, let $R$ be a semiprime ring satisfying suitable torsion restrictions, and let $T:R\longrightarrow R$ be a generalized $(m,n)$-Jordan centralizer.…

Let $G$ be a locally finite group of finite $c$-dimension $k$. Let $F(G)$ denote the Hirsh–Plotkin radical of $G$, and let $F^*(G/F(G))$ denote the generalized Fitting subgroup of…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, put $w=\beta({\bf w})$, and let $C_W(wF)$ be the centralizer of the corresponding regular element of $WF$. Let $B(w)$ be the bra…

Generalized centralizer theorem. For arbitrary $I\subseteq S$, if $\gamma\in\Pi^I$ and $s_\gamma\in W^{\perp I}{}_{\mathrm{fin}}$, then $w\cdot\gamma=\gamma$ for every $w\in Y_I$.…

Bessis–Digne–Michel's centralizer conjecture. The natural morphism

Let $M$ be a smooth manifold, let $f\in\operatorname{Diff}^\infty(M)$ be an Anosov diffeomorphism, and let $\mathcal Z^\infty(f)$ denote its $C^\infty$-centralizer. Katok--Spatzier…

Let $\mathbbm k\langle x_1,\dots,x_m\rangle$ be a free skew field over a field $\mathbbm k$, and let $a$ be a non-scalar element. The rational centralizer conjecture asserts that t…

Uniform period bound conjecture. There is a number $n(G)$ such that $p\leq n(G)$ for every $e004gamma i G$. This would provide a uniform bound for the centralizers needed by the ca…

Let $X$ be a compact homogeneous manifold, let $f_0\in\operatorname{Aff}(X)$ be a $K$-system, and suppose that $\mathcal{Z}(f_0)$ contains a hyperbolic affine diffeomorphism. If…

Let $X$ be a compact homogeneous manifold and let $f_0$ be an affine $K$-system on $X$. A smooth rank-1 factor of a centralizer means a smooth factor action obtained through a smoo…

Let $X=G/B$ be a compact homogeneous manifold, let $f_0$ be an affine diffeomorphism of $X$ admitting a $G$-invariant probability measure, and let a $K$-system mean an affine syste…

Let $M$ be a closed manifold, let $r\in \mathbb{N}\cup\\{\infty\\}$, and let $\operatorname{Diff}^r(M)$ denote the $C^r$ diffeomorphism group. The generic trivial centralizer conje…

Strengthened row-maximum conjecture. For every $i$ with $1\leq i\leq \ell$,

Let $A=U\big(\widehat{\mathfrak{gl}}(m_1+m_2|n_1+n_2)\big)$ and $B=U\big(\widehat{\mathfrak{gl}}(m_1|n_1)\big)$. Define the centralizer … Let … be the homomorphism from…

Let $B$ be an Artin group with standard generating set ${\mathbf S}$, let ${\mathbf b}\in B$ have minimal length in its conjugacy class, and let $\supp({\mathbf b})$ denote its sup…

Let $(D,\delta,d)$ be a nonassociative differential extension as above, with $C$ the distinguished subfield of $D$ and centralizer taken in $(D,\delta,d)$. Centralizer conjecture.…

Centralizer-count and derived-order conjecture. If

Let $H_3(\mathbb R)$ be the three-dimensional Heisenberg group, let $\Gamma< H_3(\mathbb R)$ be a cocompact lattice, and let … Let $\phi_{\mathbb R}$ be a uniquely ergodic, or Diop…

Let $O_q$ be the $q$-Onsager algebra, and let $\tilde G_{k+1}$, $k\in\mathbb N$, be its alternating generators. For $x\in O_q$, consider the subalgebra generated by…

Centralizer-count conjecture. The number of distinct centralizers of $G$ satisfies

Shumyatsky's conjecture. If $C_G(g)$ has finite rank for every nontrivial $g\in G$, and $G$ is not a pro-$p$ group, then $G$ has finite rank.