9 problems

Let $\mathcal{K}$ be a simplicial complex on $[m]$, let $\mathbf{k}$ be a commutative ring with unit, and define … Let…

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…

Let $M^{2n}$ be a nonsingular projective toric variety of complex dimension $n$, and let $P^n$ be its moment polytope. Flag-polytope characterization conjecture. $M^{2n}$ is a TNS-…

Let $X$ and $Y$ be nonsingular projective toric varieties of complex dimension $n$ whose moment polytopes are combinatorially equivalent. Combinatorial invariance conjecture. $X$ i…

Minkowski-relation presentation conjecture. As a vector space, $\mathcal{F}^{2k}(X)$ is generated by the elements $\{x_I\}_{I\in K}$ subject to the Minkowski relations

A simplicial sphere $K$ is strongly $B$-rigid if, whenever a simplicial sphere $L$ satisfies … then $K$ and $L$ are combinatorially equivalent. Strong B-rigidity conjecture. Every…

Indecomposability conjecture. This graded ring is indecomposable, and consequently $\mathcal {Z}_{K}$ is a prime manifold. The analogous assertion is proved in the paper for flag…

Let $K_1$ and $K_2$ be generalized homology spheres of dimension $n-1$, with $m_1$ and $m_2$ vertices, respectively. Let … The operation $\mathcal{G}$ on manifolds is the gyration,…

Diffeomorphism conjecture. $\mathcal{Z}_{K}$ is diffeomorphic to the connected sum of sphere products in Proposition 1.