17 problems

Let $G$ be an algebraic group, let $x,y\in G$, and let $\rho$ range over the irreducible rational representations of $G$. Steinberg's conjecture. The elements $x$ and $y$ are conju…

Let $f:S^1\to S^1$ be an analytic circle map, and let $\tau(f)$ denote its rotation number. Arnold conjectured that if $\tau(f)$ belongs to a set of full Lebesgue measure, then the…

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles in the same universality class, and let $h_{\gamma_i}$ be their conjugacies to the reference dynamics. Write…

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles, and let $G_{\gamma_1,\gamma_2}$ and $H_{\gamma_1,\gamma_2}$ denote the associated maps between them. Let…

Let $g$, $h$, and $h^{-1}$ be the conjugacies associated with a critical invariant circle, and let $\kappa(\cdot)$ denote their regularity. A numerical characteristic is universal…

Let $\xi$ be a braid word, and let $\zeta^*$ denote the introduced normal form of a word $\zeta$. A word $\zeta_1$ is a geodesic form of $\zeta$ when it has minimal length among wo…

Let $n\geq 0$, let $T_n$ denote the first $n$ levels of the rooted binary tree, and let $H\leq \operatorname{Aut}(T_n)$ be a group that contains an element acting transitively on…

Let $\tilde{G}$ be the relevant reductive group, let $Z(\tilde{G})$ denote its center, and let $V(\lambda_k)$, for $k=1,\ldots,l$, be the fundamental representations under consider…

Conjugacy invariance conjecture for FDSs. A shift space which is conjugated to an FDS is itself an FDS.

Uniform conjugacy invariance conjecture. Any shift which is uniformly conjugate to an SFT is itself an SFT.

Cyclic Matsumoto conjecture. The CVMT holds for a $W$-conjugacy class as long as the parabolic closure $W_{\operatorname{supp}(w)}$ of its minimal elements contains only infinite i…

Let $\alpha$ and $\beta$ be permutations in $S_n$. Write $[\alpha,\beta]=\alpha\beta\alpha^{-1}\beta^{-1}$ for their commutator, and let $\alpha^\gamma=\gamma^{-1}\alpha\gamma$ den…

Linear conjugator conjecture. In every finitely generated group, the set of Morse elements satisfies the linear conjugator property.

Premet's conjecture. There exists a nonempty Zariski open subset $V\subseteq{\mathfrak g}$ consisting of regular elements such that, for any $u,v\in V$, the Cartan subalgebras

For all $b1>1$ and $0<b2<1$, let $D_{b1,b2}$ be the set of diffeomorphisms $f\colon[0;1]\to[0;1]$ whose fixed points are exactly $0$ and $1$, with $Df(0)=b1$ and $Df(1)=b2$. C1-den…

Cyclic Matsumoto conjecture for CFC elements. The cyclic version of Matsumoto's theorem should hold for at least the CFC elements, and likely for a larger class of elements: cyclic…

Let $\operatorname{Diffeo}^+([0,+\infty))$ denote the group of orientation-preserving diffeomorphisms of $[0,+\infty)$. The conjugacy-dimension conjecture. The conjugacy dimension…