40 problems

Regular-polygon t-immersion conjecture. All $\beta(C)={n-2\choose k-1}$ solutions give rise to t-immersions, which are automatically t-embeddings.

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 the tree-level amplitudes be decomposed into terms arising from BCFW recursions, and let $\operatorname{Gr}_{4,n}$ be the Grassmannian with its cluster algebra structure. A col…

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…

Tropical fan conjecture. If the lineality space of $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ has dimension $|C|$, then the quotient of…

Consider a weighted $L^\ast$-punctured planar bipartite graph with generic edge weights $\operatorname{wt}\in\mathbb{R}_{>0}^{|\mathbf{V}^\ast|-1}$, and let $H$ be the function spe…

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 $\widetilde{\varphi}:\operatorname{Gr}(4,n)\to\operatorname{Gr}(4,n')$ be a promotion map. Promotion-map copositivity conjecture. The map $\widetilde{\varphi}$ is copositive, n…

Let $f(\mathbf{z})$ be a rational function defining a singularity hypersurface of a planar $\mathcal{N}=4$ super Yang–Mills scattering amplitude, and let…

Let $\operatorname{Gr}^{>0}_{4,n}$ be the totally positive Grassmannian, and let $f(\mathbf{z})=0$ describe a singularity hypersurface of a scattering amplitude. Scattering-amplitu…

Let $(G,\mathbf{D})$ be a tangle admitting a brushing, with $\dim\Pi_G=km$ and positive intersection number $\operatorname{IN}_m(G)>0$. Let $F_{\widehat{(G,\mathbf{D})}}$ be the as…

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…

Amplituhedron sign-characterization conjecture.

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…

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron. For a Grassmannian graph $G$, let $\Phi^\circ_G$ be the associated stratum, and let $\dim_{\mathcal{M}}(F)$ denote the dimens…

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron, and let $\mathcal{S}_{n,k}$ consist of the decorated permutations whose strata are boundary strata, together with the decorat…

Let $k,n$ satisfy $2\le k\le n-2$, let $(\tilde{\Lambda},\Lambda)$ be kinematic data as defined for the momentum amplituhedron, and let $\mathcal{M}_{n,k}(\tilde{\Lambda},\Lambda)$…

Twistor-string map diffeomorphism conjecture. The twistor-string map provides a diffeomorphism from $G_+(2,n)$ to the interior of the momentum amplituhedron. If true, this would es…

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

Let $Z$ be a totally positive matrix defining the amplituhedron $\mathcal{A}_{n,k,m}(Z)$ as the image of the positive Grassmannian under the induced map $\pi_+$. Although the defin…

Let $m^{(k)}(\alpha,\beta)$ denote the CEGM biadjoint scalar scattering amplitude for two $n$-cycles $\alpha$ and $\beta$, and let $\operatorname{Trop}_+ G(k,n)$ be the positive tr…

Forbidden-diagram characterization. For all $n$, a cell in $G_+(3,n)$ is not in $[?]$ if and only if its associated $[?]$-diagram contains one of the three forbidden $[?]$-diagrams…