15 problems

A convex Poncelet polygon is a polygon in the projective plane whose vertices lie on one conic and whose edges are tangent to another conic; convexity means convexity in an affine…

Let $T_{a,b,c,d}$ be a truly skew pentagram map, with parameters $(a,b,c,d)$ as in the paper, and let its dynamical degree be denoted by $\rho(T_{a,b,c,d})$. Khesin–Soloviev's conj…

A binomial difference operator is a difference operator whose support consists of two elements. Refactorization problems for such operators arise in the step-by-step refactorizatio…

Let $P$ be a $(k+3)$-gon in $\mathbb{RP}^k$, where $k\geq 2$, and let $J=(1,\ldots,1)$. For even $k$, let $I=(1,\ldots,1,2,1,\ldots,1)$ with the same number of $1$'s before and aft…

Let $m<d$, and consider the $(m,d+1)$-KdV flow on curves in $\mathbb{RP}^d$. A realization through subspace intersections is a construction in which an $(m-1)$-dimensional space is…

A KdV-type equation is an equation arising in the KdV hierarchy, and a pentagram-type map is a discrete map on polygons whose continuous limit produces an evolution equation. Discr…

Let $T_{I,J}$ be a generalized pentagram map associated with two multi-indices $I$ and $J$. A map is Hamiltonian here if it preserves a certain nontrivial Poisson structure. Khesin…

Let $(v_i)$ be a twisted polygon in $b2b2\text{?}b2b2$ and let $I$ be a multi-index defining a generalized pentagram map $T_{I,J}$. The map $T_{I,I}$ is the diagonal case in which…

Density and finite-to-one conjecture. The map $\pi_1$ has dense image. Restricted to twisted configurations, the map $\pi_1$ is finite-to-one. This predicts both generic realizabil…

Let $S$ be a horizontal $Y$-pin with $a_2=b_2=0$, $0=a_1<b_1$, $p=c_2$ and $q=d_2$ relatively prime. Let $(P_{i,j})_{i,j\in\mathbb{Z}}$ be a $Y$-mesh of type $S$ and dimension…

Let $P\subset\mathbb{P}^2$ be an axis-aligned $2n$-gon. Let $T$ denote the pentagram map, and let … be its point of collapse. Let $\mathscr C(P)$ denote the center of mass of $P$.…

Let $T_{\mathcal I,\mathcal J}$ be a universal pentagram map on $d$-tuples of twisted $n$-gons in ${\mathbb {RP}}^d$, determined by matrices $\mathcal I$ and $\mathcal J$. Let…

Let $T_*$ denote an integrable pentagram map on twisted $n$-gons in ${\mathbb {CP}}^d$, with parameters $n$, $m$, and $d$. A discrete map is included into a flow if it occurs as th…

Non-integrability conjecture. This generalized pentagram map is non-integrable. The conjecture is supported by computer experiments, alongside examples of both integrable and non-i…

Khesin–Soloviev higher-dimensional pentagram Devron conjecture. If $B=F^{n-3}(A)$, then $B_1,B_3,\ldots,B_{2n-1}$ are coplanar, and $B_2,B_4,\ldots,B_{2n}$ are also coplanar.