241 problems

A framework $(\mathcal{G},p)$ in $\mathbb{R}^2$ is infinitesimally angle rigid if its infinitesimal angle constraints determine its realizations up to the relevant trivial motions,…

Let $G=(V,P)$ be a graph, and let $\mathcal{P}$ denote its set of prescribed pairs. A subgraph is globally rigid in $\mathbb{R}$ if its realization in the real line is uniquely det…

Edge-bound conjecture. One has

Symmetry-count sufficiency conjecture. For all such groups $\Gamma$, the counts in Corollary are sufficient for $(G,\chi,p)$ to be $\Gamma$-symmetric isostatic.

Let $G$ be a minimally rigid graph on $n$ vertices, and let $\mathbf{c}_{2}(G)$ be the graph invariant described in the source's introduction. Jackson–Owen conjecture. One has … Th…

Redundant symmetry-rigidity conjecture. For every such $\Gamma$ and $e$, $(G({\cal S})-e-e^*,*)$ is $\Gamma$-rigid in $\mathbb R^{k+1}$.

Let $d\geq 1$ and let $P_{n,d}$ be the generalized path graph, an $n$-vertex minimally $d$-rigid graph. For a graph $G$, let $a_d(G)$ denote its $d$-dimensional algebraic connectiv…

Pinned symmetric Laman conjecture. The standard Laman-type conditions for $G$, together with the additional necessary conditions concerning the number of fixed joints and bars, sho…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group whose factors are all non-compact, let $\Gamma<G$ be an irreducible lattice, and let $\rho$ be an affine action of…

Let $G$ be a semi-simple Lie group with no compact factors and no simple factors isomorphic to $SO(1,n)$ or $SU(1,n)$, and let $\Gamma<G$ be a lattice. A generalized quasi-affine a…

The 3-connected Laman graph non-solvability conjecture. If a generic Laman graph is $3$-connected, then its constraint equations are not solvable by radicals (not-RS).

Refinement conjecture. If $(G,\rho)$ is refined to $(G',\rho)$ without adding vertices, then $(G',\rho)$ is good too.

Let $G$ be a plane graph. A plane graph is generically rigid if its generic realizations are infinitesimally rigid, and it can be straightened as a pseudo-triangulation if its embe…

Let $G=(V,E)$ be a graph, and let ${\mathcal V}(G)$ denote its slope variety. For each circuit of the generic rigidity matroid on $V$, consider the corresponding slope relation amo…

Global smooth classification conjecture. All Anosov actions of such lattices are smoothly equivalent to one of the above algebraic actions.

Six-connectivity conjecture. Every 6-connected $K_4$-covered graph is rigid in $\mathbb{R}^3$.

Kiraly–Tanigawa's body-pin conjecture. The graph $G_H$ is rigid in $\mathbb{R}^3$ if and only if, for every partition $\mathcal{P}=\{P_1,P_2,\dots,P_t\}$ of $W$,

Let $S$ be a closed piecewise-linear surface, and let $(G_S^\star,\boldsymbol p)$ be its coned framework: the framework has the vertices and edges of $S$, together with a cone vert…

Let $S(t)$ be a differentiable one-parameter family of oriented piecewise-linear surfaces. For each facet $F$, let $n_F$ be its unit normal and $V_F$ its volume, and define the dih…

Let $S$ be a closed piecewise-linear surface, with facets indexed by $i$, unit facet normals $n_i$, facet volumes $V_i$, and dihedral angles $\theta_{FG}:=\angle(n_F,n_G)$. Conside…

Let $d\geq 3$, and let $p:[n]\to\mathbb{R}^d$ satisfy … for all $i\in[n]$ and … Assume that the image of $p$ has size at least $3$. Lew–Noy–Plaza–Eberhardt's conjecture. The second…

Let $g\geq 2$, let $S_g$ be the closed orientable surface of genus $g$, and let a simple graph be called rigid in $S_g$ when it has a well-positioned realisation whose associated g…

Jackson–Tanigawa's rigidity conjecture. For $d\geq 3$ and $\mathcal{X}=\{K_{d+2},K_{d+2,d+2}\}$, $\mathcal{R}_n^d$ is the unique maximal $\mathcal{X}$-matroid, and…

Jackson–Tanigawa's hyperconnectivity conjecture. For $d\geq 1$ and $\mathcal{X}=\{K_{d+2},K_{d+1,d+1}\}$, $\mathcal{H}_n^d$ is the unique maximal $\mathcal{X}$-matroid, and…

Let $G_{n,k}$ be the uniform random $n$-vertex $k$-regular graph. Random-regular rigidity conjecture. For every $d\geq 2$ and $k\geq 2d$, the random graph $G_{n,k}$ is a.a.s. $d$-r…