14 problems

Let $C$ and $C'$ be arbitrary complex antisymmetric matrices, and let $W$ be a matrix giving their simultaneous canonical forms. In the fully diagonalizable case, the simple forms…

Consider the truncated-support subtraction-game positions and the options described immediately before the conjecture: the source identifies a specific collection of dominated opti…

Let $\chi_n$ be the electroid space, let closed electroid varieties be its closed electroid subvarieties, and let the electroids be partially ordered by their poset of electroids.…

Converse parametrization conjecture. If a Brunovsky transformation $(T,F,G)$ transforms the staircase linear system into Brunovsky form, then there exists an observation matrix $C$…

Parametrization completeness conjecture. The set of linearly parameterized transformations constructed in Sections and contains all transformations that map the staircase system to…

The common-ground conjecture. The pair $\{E,F\}$ is transformable into strong standard canonical form, and the Jordan normal form of its constant nilpotent matrix $N$ consists exac…

Let $E,F:\mathcal I\rightarrow\mathbb{R}^{m\times m}$ be sufficiently smooth, and let $\{E,F\}$ be a regular pair with index $\mu\geq 2$. Write $r$ for the dimension of the differe…

Let $\mathcal{A}_{n,k,m}(Z)$ be an amplituhedron and let $\{Z_S\}_{S\in\mathcal{C}}$ be a tiling of it. Canonical form from tilings. The canonical form of the amplituhedron is obta…

Let $(X,X_{\geq 0})$ be a positive geometry. A subdivision is a collection of positive geometries $(X_1,X_{1,\geq 0}),\ldots,(X_r,X_{r,\geq 0})$ with pairwise disjoint interiors an…

For integers satisfying $k e 0$ and $k+m e n$, let $Z$ be a totally positive matrix and let $mathcal{A}{n,k,m}(Z)$ be the image of $mathop{rm Gr}+(k,n)$ under the induced map. A po…

Let ${\mathcal A}$ be an Amplituhedron, let $Y\in\operatorname{Int}({\mathcal A})$ be an interior point, and let $d\operatorname{Vol}_{k,m}(Y,W)$ be the natural form on pairs consi…

Let ${\mathcal A}(k,n,m;l^L)$ be an Amplituhedron, let $\Delta^D$ have positive coordinates $X_1,\ldots,X_D$, and let … be a morphism from positive coordinates to the Amplituhedron…

Let $p$ be a general binary form of even degree $2s$, and let $\alpha_j,\beta_j\in\mathbb{C}$ for $1\leq j\leq s+1$. Canonical representation conjecture. The form $p$ can be writte…

Let $Q$ be a finite connected quandle with elements $1,2,\ldots,n$ naturally ordered, profile $\{1,\ell_1,\ell_2,\ldots,\ell_k\}$, and $1\leq\ell_1\leq\ell_2\leq\cdots\leq\ell_k$.…