Gradient Flows and Ball Topology of Amplituhedra

Definitions and mathematical statement

The supplied record concerns the amplituhedron $A_{n,k,m}(Z)$, but does not define it or specify the admissible parameters. The following is a conventional candidate definition, included conditionally rather than asserted to be the exact convention of the source. Let $n,k,m$ be nonnegative integers with $k+m\leq n$. Write $\operatorname{Gr}(k,n)$ for the Grassmannian of $k$-dimensional linear subspaces of ${\mathbb{R}}^n$, and let $\operatorname{Gr}_{\geq0}(k,n)$ consist of the row spaces of rank-$k$ matrices $C\in{\mathbb{R}}^{k\times n}$ whose ordered maximal minors are all nonnegative. In a column convention for the external data, let $Z\in{\mathbb{R}}^{(k+m)\times n}$ have every ordered maximal minor strictly positive. The conventional amplituhedron is \[ A_{n,k,m}(Z)=\left\{\operatorname{rowspan}(CZ^{\mathsf T}):\operatorname{rowspan}(C)\in\operatorname{Gr}_{\geq0}(k,n)\right\}\subseteq\operatorname{Gr}(k,k+m). \] The intended parameter range, any parity restrictions on $m$, and the source's precise positivity and matrix conventions require confirmation.

To explain the flow terminology, let $M$ be a smooth manifold containing the amplituhedron, let $g$ be a Riemannian metric on $M$, and let $f:M\to{\mathbb{R}}$ be smooth. A negative gradient flow satisfies \[ \frac{{\,\mathrm{d}}}{{\,\mathrm{d}} t}\Phi_t(x)=-\operatorname{grad}_g f\left(\Phi_t(x)\right),\qquad \Phi_0(x)=x. \] Forward preservation of $A=A_{n,k,m}(Z)$ means that the flow is defined on $A$ for every $t\geq0$ and \[ \Phi_t(A)\subseteq A\qquad(t\geq0). \] This definition is an explicit possible interpretation: the source does not specify the ambient manifold, metric, potential, sign convention, or whether preservation is forward or two-sided.

Requested classification and existence problem. For fixed admissible parameters $(n,k,m)$, classify the gradient flows preserving $A_{n,k,m}(Z)$. Determine whether, for every admissible positive $Z$, there exists such a flow with additional properties sufficient to establish a homeomorphism \[ A_{n,k,m}(Z)\cong\overline B^{\,d},\qquad \overline B^{\,d}=\left\{x\in{\mathbb{R}}^d:\sum_{i=1}^{d}x_i^2\leq1\right\}, \] for the intended nonnegative integer dimension $d$. The record does not specify $d$, the equivalence relation for the classification, or the additional dynamical hypotheses needed for this topological conclusion. Preservation alone is insufficient: the constant potential gives the identity flow, which preserves every subset without imposing closed-ball topology. Consequently, this is a conditional statement schema requiring clarification, rather than a fully specified conjecture.

Short English statement

Classify gradient flows preserving the amplituhedron $A_{n,k,m}(Z)$, and determine whether every admissible positive $Z$ admits a suitable flow establishing that the amplituhedron is homeomorphic to a closed ball. The supplied record leaves the admissible parameters, precise flow class, classification equivalence, and sufficient dynamical conditions unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004941$, title Gradient Flows and Ball Topology of Amplituhedra, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied primary-source citation: Mini-Workshop: Scattering Amplitudes, Cluster Algebras, and Positive Geometries (2022), Oberwolfach Reports, pp. 3101--3133, DOI 10.4171/owr/2021/57. The extracted problem identifies p. 3127. https://doi.org/10.4171/owr/2021/57. The registry separately records proposed year 2021 and the tag owr-year:2021; the supplied record does not reconcile these with the citation's 2022 date.
• Supplied literature reference: Karp--Williams, Gradient Flows and Adjoint Orbits. https://doi.org/10.1007/s00220-022-04540-5.
• Supplied literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2021/57.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the problem open and states that gradient-flow methods establish closed-ball topology for several important amplituhedron families, while the requested classification and suitable-flow existence for every positive $Z$ remain open. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports direct checking of the official report. These are supplied provenance assertions; no independent source visit or mathematical verification is claimed here.