70 problems

Let $V(3,n)_{\geq 0}$ be the totally nonnegative part of the ABCT variety, and let the momentum amplituhedron be the positive geometry arising in the corresponding scattering-ampli…

Let $V(k,n)$ be the Zariski closure of the image of the rational Veronese map from $\operatorname{Gr}(2,n)$ to $\operatorname{Gr}(k,n)$, and let … denote its analytic closure. Lam'…

Let $P$ be a regular polypol, meaning that its boundary contains no singular points of its ambient variety $Y$ except for the vertices and that…

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.…

For a configuration of points on a twisted cubic in $\mathbb{P}^3$, the two described degenerations are: two points collide, or the twisted cubic degenerates to a conic and a line…

Let $G$ be a finite graph, let $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote the variables of its flat space wavefunction $\psi_G$, and let $\Omega_G$ be the canonical form of the c…

Let $n,k,m$ be positive integers with $k+m\leq n$, and let $Z$ be a generic $n\times(k+m)$ matrix all of whose minors are positive. Write $\mathcal{A}_{n,k,m}(Z)$ for the image of…

Amplituhedron canonical-form conjecture. When expressed in terms of lines in $\mathbb{P}^3$, the rational function is conjectured to be the canonical form of the positive geometry…

Let $A_{n,k,m}$ be the amplituhedron associated to a positive map $Z:{\mathbb{R}}^n\to{\mathbb{R}}^{k+m}$. A positive geometry is a pair $(X,X_{\geq 0})$ equipped with a canonical…

Let $\operatorname{Gr}(k,n)_{\geq 0}$ be the totally nonnegative Grassmannian, let $Z:{\mathbb{R}}^n\to{\mathbb{R}}^{k+m}$ be positive, and let … be the associated amplituhedron. T…

Let $d538_{k,m,n}(Z)\subset\operatorname{Gr}(k,k+m)$ be the amplituhedron associated with positive external data $Z$. For odd $m$, impose positivity of the displayed brackets invol…

Let $d538_{k,4,n}(Z)$ denote the tree amplituhedron associated with positive data $Z$. A positive geometry is a complex algebraic variety defined over $d53f$, together with a semia…

The tree-level amplituhedron ${\cal A}_{k,n,m}$ is a semi-algebraic subset of the real points of the Grassmannian $X={\rm Gr}(k,k+m)$. It is known to be a positive geometry in the…

Let $U(\Sigma)$ be the upper cluster algebra and let $U(\Sigma)^+$ denote its positive Laurent polynomials in a cluster chart. Let $\mathcal{X}(\mathbb{Z}^t)$ be the tropical integ…

A positive hexahedron $S$ is obtained by intersecting the positive Grassmannian $\mathrm{Gr}_{\geq 0}(2,4)$ with two additional linear half-spaces. An adjoint polynomial is a polyn…

Fix $Z\in\mathbb{R}^{5\times 4}$ with all ordered $4\times4$ minors positive, and let ${\cal A}_{5,2,2}(Z)\subset\operatorname{Gr}_{\mathbb{R}}(2,4)$ be the image of the nonnegativ…

Fix a $2k\times(2k+2)$ matrix $\Lambda$. The orthogonal momentum, or ABJM, amplituhedron $\mathcal{O}_k(\Lambda)$ is the image of $\mathrm{OG}^{\geq}_{k,2k}$ under the map…

Let $k\geq 3$. Let $\Lambda$ be a $2k\times(k+2)$ matrix with all positive maximal minors, and let $\mathcal{O}_k(\Lambda)$ be the ABJM amplituhedron. For each…

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

Chirotropical Grassmannian conjecture. Under these assumptions, the equality of sets

Let $\mathcal{A}_{k,n,4,L}$ denote the loop Amplituhedron, a semialgebraic set in … A canonical form is the differential form associated with a positive geometry whose logarithmic…

Cyclically symmetric positivity conjecture. The cyclically symmetric momentum amplituhedron $\mathcal{M}_{\Lambda^{\operatorname{cs}},\tilde\Lambda^{\operatorname{cs}}}$ is Mandels…

Let $\operatorname{Gr}(2,n)_{>0}$ be the positive Grassmannian, and let the scattering map of the twistor construction map it to the momentum amplituhedron. Denote the interior of…

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