6 problems

Let $n\geq 3$ and let $\mathrm{r}=(r_1,\dots,r_n)\in\mathbb{R}^n$ have positive coordinates. Let $X(\mathrm{r})$ be the space of simple $n$-gons $(z_1,\dots,z_n)$ with…

Global cone conjecture for moduli spaces of simple polygons. ${\mathcal M}(n)$ is homeomorphic to $C(\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle)$. A local version holds in a neighbor…

Let $S(n)$ be the connected component of the space of simple labeled $n$-gons modulo oriented similarity consisting of positively oriented polygons. As an open subset of…

Gromov-width conjecture. The Gromov width of $\mathcal{M}(r)$ is

Let $X_{a,b}(\ell)$ be a big polygon space associated with a length vector $\ell$, let $T$ be the acting torus, and let $\mu(\ell)$ denote the invariant introduced in the paper. Th…

Nonexistence conjecture. For an arbitrary $n$ with $n\geq n_0$, where $n_0$ is sufficiently large, the isotopy of Theorem 1 is not valid, even without the additional requirements t…