3 problems

$c_1$-cohomological rigidity conjecture. If there exists a $c_1$-preserving graded ring isomorphism between the integral cohomology rings of $X$ and $Y$, then $X$ and $Y$ are isomo…

A Bott tower of height $n$ is an iterated $\mathbb C P^1$-bundle whose stages are Bott manifolds; each Bott manifold is a closed manifold of dimension $2n$. Strong cohomological ri…

Katok–Herman conjecture. The only examples, up to smooth diffeomorphism and conjugacy, of $CR(M)$ diffeomorphisms are Diophantine translations in tori.