10 problems

A closed manifold $M$ is almost flat if, for every $\epsilon>0$, it admits a Riemannian metric $g_\epsilon$ satisfying … where $K_{g_\epsilon}$ denotes sectional curvature. Farrell…

Let $M$ be a compact differentiable manifold and let $f\colon M\to M$ be an Anosov diffeomorphism, meaning that the tangent bundle admits a continuous invariant splitting … with…

An Anosov diffeomorphism is a diffeomorphism of a compact manifold with a uniformly hyperbolic tangent-bundle splitting. An infranilmanifold is a compact manifold finitely covered…

Franks–Manning conjecture. Smooth Anosov diffeomorphisms can exist only on infranilmanifolds.

For $K\rrbracketin\mathbb{R}$ and $N\in\mathbb{N}$, let an $\operatorname{RCD}(K,N+\varepsilon)$ space mean a metric-measure space satisfying the RCD condition with parameters…

Almost-flat RCD+CBA conjecture. For any $1<N<\infty$, there exists $\varepsilon=\varepsilon(N)$ with the stated property.

Let $p_i\colon E_i\to X$, $i=1,2$, be Anosov bundles. Their fibers are assumed to be infranilmanifolds. Infranilmanifold-fiber conjecture. Addendum remains true in this more genera…

Infranilmanifold global rigidity conjecture. All irreducible Anosov genuine $\mathbb{Z}^r$-actions on compact infranilmanifolds should be $C^\infty$-conjugate to actions by affine…

Let $M$ be an infranilmanifold, let $\alpha_0$ be a totally non-symplectic $\mathbb{Z}^k$ action by automorphisms of $M$, and let $\alpha$ be an action with homotopy data…

Let $N$ be a simply connected nilpotent Lie group, let $A$ be a group of affine transformations acting freely with a finite-index translation subgroup that is a lattice in $N$, and…