67 problems

Let $\mathcal{A}_{n,k,2}$ be the $m=2$ amplituhedron, and let $\operatorname{Trop}^+{\rm Gr}(k+1,n)$ denote the positive tropical Grassmannian. A regular positroidal subdivision is…

Let $\Gamma_{L^\ast}$ be a graph in the class of loop BCFW graphs $\mathbf{\Gamma}^{\mathtt{BCFW}}_{k,n;L^\ast}$, and let … be the corresponding branch of the loop BCFW recursion.…

Let $\mathcal{M}_{k,n;L}$ and $\mathcal{A}_{k-2,n;L}$ denote respectively the $m=4$ momentum and momentum-twistor amplituhedra for $n$ particles, helicity $k$, and loop order…

The amplituhedron is the geometric space arising from the positive Grassmannian in the scattering-amplitude construction, and the BCFW recurrence relations are the Britto--Cachazo-…

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…

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 $C_{k,m,n}(Z)$ be the exterior cyclic polytope constructed from positive data $Z$, and let its $f$-vector record the numbers of faces in each dimension. Constant f-vector conje…

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…

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…

Amplituhedron sign-characterization conjecture.

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

HZ's positive-region integral conjecture. The scattering amplitude should be given by an integral over a certain $(2n-4)$-dimensional slice of this positive region.

Let $\mathcal{A}_{n,k,m}$ be the amplituhedron, and let a tiling mean a tiling of this geometric object. For even $m$, consider the number of plane partitions fitting inside a…

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 the image of the positive Grassmannian $\operatorname{Gr}_{k,n}^{\scriptscriptstyle\geq 0}$ under the amplituhedron map … A tile is a smaller region…

Lam's conjecture. This condition is necessary for $\widetilde{Z}$ to be well-defined on $\operatorname{Gr}_{\geq 0}(k,n)$.

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 $\mathcal{M}^+_{0,2k}$ be the positive moduli space, let $\overline{\mathcal{M}}^+_{0,2k}$ be its compactification, and let $\mathcal{O}_k$ be the orthogonal momentum amplituhe…

Let $\mathcal{F}_k$ denote the set of orthogonal Grassmannian forests of type $\underline{k}$, and let $\Phi_\Gamma^\circ$ be the stratum associated with $\Gamma\in\mathcal{F}_k$.…

Positivity conjecture. The planar Mandelstam variables are positive for all $Y\in\mathcal{O}_k$. Positivity of these variables would provide a key positivity property of the orthog…

Let $\mathcal{O}_k$ be the image of the positive orthogonal Grassmannian under the map determined by a positive $(2k)\times(k+2)$ matrix $\Lambda$ whose entries lie on the moment c…

Let $Z$ be an $n\times(k+m)$ matrix with positive maximal minors, let $\mathcal{A}_{n,k,m}(Z)$ be its amplituhedron, and let a positroid tile be a tile in a positroid tiling of thi…