17 problems

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.…

Realizability conjecture for orthopositroids. Every orthopositroid $\mathscr{M}$ of type $(k,n)$ is realizable: there exists $X\in\operatorname{OGr}_{+}(k,n)$ such that…

Yang–Mills scattering form conjecture. The $(2n-4)$-form $\Upsilon_n$ coincides with the $\mathcal{N}=4$ SYM scattering form defined by HZ.

Positive geometry conjecture. The closure

Let $\Delta$ be the face complex of an $n$-gon. A binary geometry for $\Delta$ is a binary geometry whose underlying flag complex is $\Delta$. Binary geometry existence conjecture.…

For ${\mathcal{N}} \in Q_{n,2}$, let $A_{\mathcal{N}}:=A_{n,k,2}\cap {\mathring{\Pi}}_{\mathcal{N}}$, where $A_{n,k,2}$ is the $m=2$ amplituhedron and…

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 $Z_S$ be a tile of the amplituhedron $\mathcal{A}_{n,k,m}(Z)$, and let $\tilde{\Omega}(Z_S)$ denote its candidate canonical form, defined by pulling back the canonical form of…

Let $\mathcal{A}_{n,k,m}(Z)$ be the amplituhedron in $\operatorname{Gr}_{k,k+m}(\mathbb{C})$. The amplituhedron positive geometry conjecture. The pair … should be a positive geomet…

Let $\mathcal{A}$ be a cluster algebra with a good basis $\mathfrak{B}$, and let $\mathfrak{B}_{ir}$ be the subset of indecomposable real elements, where an element $b$ is real if…

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…

Let $Z:{\mathbb R}^n\to{\mathbb R}^{k+m}$ be a linear map with $n\geq k+m$, and let $P=Z({\mathrm{Gr}}(k,n)_{\geq 0})$ be the resulting Grassmann polytope when $Z$ is well-defined…

Let $Y$ be a cluster variety that is locally acyclic and of full rank. Write $Y_{>0}$ for its positive part and $\Omega(Y_{>0})$ for its natural top form. The simplex-like compactif…

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 $\operatorname{OG}_{\ge0}(n,2n)$ be the positive orthogonal Grassmannian, with cell decomposition … where the cells are indexed by matchings $\tau$ on $[2n]$, and the cell inde…

Polynomial numerator conjecture. There exists a polynomial $H(S,A,B)$ in the entries of $S,A,B$ such that

Let ${\mathcal A}$ be an Amplituhedron embedded in an irreducible variety $X$, and let $\mathcal Y\in X$ lie away from the boundary of ${\mathcal A}$. Dual Amplituhedron volume con…