22 problems

Let $\operatorname{Gr}_{k,n}^{\scriptscriptstyle\geq 0}$ be the positive Grassmannian, let its BCFW recurrence determine a collection of positroid cells, and let … be the amplituhe…

Let $k\geq 1$ and $n\geq k+4$. Let $\mathcal{A}_{n,k,4}$ be the amplituhedron, and let $\mathcal{BCFW}_{n,k}$ denote the collection of $4k$-dimensional BCFW positroid cells in the…

The amplituhedron is a geometric region in momentum-twistor space whose canonical form is intended to produce scattering amplitudes at tree level and loop integrands at loop level.…

Let $(G,\mathbf{D})$ be a plabic tangle of dimension at most $km$ admitting a brushing. Let $F_{\widehat{(G,\mathbf{D})}}$ be its associated space. Higher-intersection positivity c…

Let the tiles of the amplituhedron be the images of the relevant positroid cells under the amplituhedron map, and call cluster variables compatible when they occur together in a cl…

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…

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…

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…

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 $\mathcal{A}_{n,k,m}$ be the tree amplituhedron, a subspace of the Grassmannian $\operatorname{Gr}(k,k+m)$. For a point of $\operatorname{Gr}(k,k+m)$, consider the associated c…

Let $m$ be the twistor dimension, let $k=n-m$, and let $\mathscr{H}_{n,n-m}^{(f)}$ be a maximal generalized amplituhedron-like geometry. Let $H_{n,n-m}^{(f)}$ be its canonical form…

Let $k=n-4$. Let $H_{n,n-4,l}^{(f;f+2,\ldots,f+2,f,\ldots,f)}$ be the canonical form of the loop amplituhedron-like geometry with $l'$ loop variables of maximal flipping number…

Let $\mathscr{H}_{n,n-4}^{(f)}$ be a maximal amplituhedron-like geometry, let $H_{n,n-4}^{(f)}$ be its oriented canonical form, and let $A_{n,f}$ denote the corresponding superampl…

Let $\braket{ijkl}$ denote a configuration of brackets with positive proper boundaries and flipping number $f$. Let $\tilde Z$ be a positive configuration and let…

Let the squared amplituhedron be the union of all amplituhedron-like geometries, and let its oriented canonical form be the corresponding canonical form. Squared amplituhedron conj…

Let $(\mathcal{P},[n])$ be a Wilson loop diagram, with associated positroid cells of dimension $3|\mathcal{P}|$, and let the Amplituhedron parametrize positroid cells of dimension…

The Amplituhedron is the geometric object associated with the positive Grassmannian in the scattering-amplitude framework. Orientability conjecture. The Amplituhedron is orientable…

The amplituhedron is a positive geometric object in momentum-twistor space, with loop configurations subject to positivity and mutual-positivity conditions. Its canonical forms pro…

He–Kojima–Zhang conjecture. The positive region in spinor-helicity space should be defined by proper sign flips for both $\lambda_i$ and $\tilde\lambda_i$, together with

Tree amplituhedron membership conjecture. $Y$ is in the amplituhedron if and only if the obvious boundaries