4 problems

Let $L$ be an irreducible hyperbolic toral automorphism, and let $f$ be a C^ diffeomorphism of the torus that is $C^1$-conjugate to $L$. Gogolev's smooth rigidity conjecture. The…

Kanai's conjecture. If the distribution $\mathbb{H}$ is $\mathcal{C}^1$, then $(M,g)$ is locally symmetric.

A nilflow is a flow generated by a one-parameter subgroup on a nilmanifold; a flow is geometrically rigid when continuous conjugacies within the relevant smooth class are smooth. C…

Khanin's rigidity conjecture. If $T_1$ and $T_2$ are topologically conjugate, then the conjugacy is $C^\infty$ smooth; it may suffice to require the Diophantine condition for only…