9 problems

Spectral rigidity conjecture. Spectral rigidity should hold for all Anosov metrics in any dimension.

Let $\rho_0$ and $\rho_1$ be two distinct rotation numbers in $(0,1/2]$. For ellipses $\mathscr E$ and $\mathscr E'$, let $\beta_{\mathscr E}(\rho)$ denote the Mather beta function…

Generic length-spectrum injectivity conjecture. There is a Zariski closed proper subset $E_d\subset {\mathcal M}_d$ defined over $\mathbb R$ such that, for every $[f]\notin E_d$, i…

Multiplier-spectrum injectivity conjecture. If $\tau_d(f)=\tau_d(g)$, then either

Length-spectrum rigidity conjecture. If $f$ and $g$ have the same length spectrum, then

White-noise semigroup variance conjecture. One has

Let $S$ be the underlying surface and let $\mathrm{Flat}(S)$ denote the space of flat metrics on $S$. For $\varphi\in\mathrm{Flat}(S)$, let $L_{\varphi}$ be its Liouville geodesic…

Let $n$ be a positive integer, and consider the $n$-dimensional ball as a domain in $\mathbb{R}^n$. Polterovich–Sher's conjecture. The $n$-dimensional ball is uniquely determined b…

Let $F_N$ be a free group, let $H\leq\operatorname{Aut}(F_N)$ be cyclic, and let $g\in F_N$. The orbit of $g$ is $Hg\subseteq F_N$. Non-rigidity conjecture for cyclic automorphism…