7 problems

Let two toric manifolds, equivalently smooth toric varieties, be given. Their integral cohomology rings are isomorphic if and only if the manifolds are diffeomorphic. Masuda–Suh co…

$c_1$-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…

Weak Katok–Herman conjecture.

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

Let $X(A)$ and $X(A')$ be Bott manifolds of complex dimension $n$. Their integral cohomology rings are graded rings of the form $H^*(X(A))$ and $H^*(X(A'))$. Bott manifold cohomolo…

Strong cohomological rigidity conjecture for Bott manifolds. Any graded ring isomorphism between the cohomology rings (with integer coefficients) of two Bott manifolds is induced f…