9 problems

Let $M$ be a compact surface of negative curvature, let $-\Delta_g\psi_n=E_n\psi_n$ be Laplace–Beltrami eigenfunctions, let $\nu_g$ be the normalized Liouville measure on the unit…

Let $M$ be a complete Riemannian manifold of dimension $n$ without conjugate points, with Green subbundles $X^s(\theta),X^u(\theta)$ depending continuously on $\theta\in SM$ and cu…

Let $(M,g_0)$ be a smooth closed Riemannian manifold with Anosov geodesic flow, and let $I^{g_0}_2$ be the X-ray transform on symmetric two-tensors, defined by … Here $D^*$ is the…

Let $(M,g)$ be a smooth Anosov Riemannian manifold, let $\mathcal{E}$ be a Hermitian vector bundle over $M$, and let $u$ be a solution of the twisted cohomological equation … Here…

Smale–Kalinin–Spatzier algebricity conjecture. Every Anosov diffeomorphism and every Anosov action of $\mathbb{R}^k$ for $k\geq 2$ should be conjugate, up to finite covers and repa…

Distance equality conjecture. The distances $d_T$ and $d_F$ coincide on these isometry classes. The inequality $d_T\leq d_F$ is established, and equality is known on Teichmüller sp…

Hypoelliptic damping conjecture. The second part of the theorem for the elliptically perturbed operator $P_\epsilon$ holds with $\widetilde P_\epsilon$ in place of $P_\epsilon$. Th…

Let $M$ be a 3-manifold and let $f:M\to M$ be a partially hyperbolic diffeomorphism with invariant bundles $E^u$, $E^c$, and $E^s$. An embedded torus is periodic if $f^k(S)=S$ for…

Let $(M,g)$ be a negatively curved manifold whose unit cotangent bundle $S^*M$ has Anosov geodesic flow $(g^t)$. Let…