16 problems

Schwartz's conjecture. The real pentagram map might be a rare example of a Liouville–Arnold discrete integrable system.

Generalized Glick conjecture. Let $k\geq 2$. If $n=3k+1$, the point $\widehat P_{\infty}$ is a fixed point for the projective action of $G_{P,k,k}$. If $n=3k+2$, the point…

Pre-compactness conjecture. The sequence $\{P^{\ell}\}$ is pre-compact modulo affine transformations. That is, this sequence has a convergent subsequence which converges to another…

Let $m\geq 1$, let $\tilde{f}:\mathbb{Z}^2\rightarrow {\mathbb{R}\mathrm{P}}^2$ be a $2m$-closed pentagram lattice map, equivalently let…

Vanishing-density conjecture. The dynamical domain has asymptotic density zero over the tower of finite fields:

The generalized pentagram map is considered in the case of projective space $\mathbb{RP}^{m-1}$. A scaling means a modification of the map by a scaling factor chosen so that the re…

Integrability conjecture. The map $T_{n,k}$ acting on ${\cal C}(n,k)$ is a discrete totally integrable system.

Uniqueness conjecture. For any $(n,k)$, the invariant $Z$ is uniquely maximized, and has a unique critical point, at the point representing the logarithmic pentagram spiral.

Khesin–Soloviev scaling invariance conjecture. This scaling should leave the pentagram map invariant. This conjecture was proved for $s=1$ and checked computationally for $s=2$; it…

Let $T$ be the higher pentagram map on generic twisted $n$-gons in $\mathbb{CP}^d$, and let $a_{j,k}$, $0\leq j\leq n-1$, $1\leq k\leq d$, be the coordinates defined by the normali…

Singularity confinement conjecture. Singularity confinement holds generically on $X_S$ unless

The pentagram map sends the $i$-th vertex of a polygon to the intersection of the diagonals $(i-1,i+1)$ and $(i,i+2)$; it is invariant under projective transformations. On the spac…

Let $k$-AGD flow be the flow in $m$ dimensions associated with the corresponding AGD Hamiltonian, and let $\mathbb{RP}^m$ be real projective $m$-space. A $k$-AGD discretization con…

No-other-relations conjecture. There are no other relations between the monodromy invariants that hold identically on ${\cal C}_n$.

Let the pentagram invariants define complex algebraic varieties, and suppose these varieties are complex tori after a suitable compactification. Equip the compactified tori with th…

Let the pentagram map act on the spaces of twisted $n$-gons, and let the pentagram invariants be the polynomial invariants described in the paper. Completeness conjecture. The pent…