Geometry and Order of Caustics

Definitions and mathematical statement

The supplied record asks: What are caustics in the oscillatory-integral setting, and how can their geometry and order be determined? It describes caustics as envelopes of optical rays or as degeneracies of stationary points of a phase, associated with failure of uniform $L^\infty$ bounds. This is a research question whose mathematical setting is not specified sufficiently to give a unique quantified statement.

To explain the phase terminology only, let $X\subseteq{\mathbb{R}}^d$ and $\Theta\subseteq{\mathbb{R}}^m$ be open sets, with $d,m\geq1$, and let $\phi\in C^2(X\times\Theta;{\mathbb{R}})$. These are auxiliary conventions, not hypotheses supplied for the intended problem. A stationary point in the integration variables is a pair $(x,\theta)$ satisfying \[ \nabla_\theta\phi(x,\theta)=0. \] It is degenerate when \[ \det D^2_{\theta\theta}\phi(x,\theta)=0. \] The corresponding set of parameter values is \[ \mathcal K_\phi=\left\{x\in X:\ \exists\theta\in\Theta,\ \nabla_\theta\phi(x,\theta)=0,\ \det D^2_{\theta\theta}\phi(x,\theta)=0\right\}. \] This set expresses a possible local phase description of a caustic. Identifying it with an envelope of optical rays requires additional geometric hypotheses and a specified ray family, neither of which is supplied.

Statement of the supplied research target. Determine the geometry and order of caustics in the intended oscillatory-integral or optical-ray setting, and characterize the associated failure of the intended uniform $L^\infty$ estimates. The source does not specify the family of integrals, phase class, amplitude, parameter regime, ray dynamics, domains, boundary conditions, or meaning of "order." It also does not identify which bound fails or what its uniformity means. Degeneracy alone does not imply failure of every uniform $L^\infty$ bound; the relevant normalization, amplitude, and limiting regime must be specified. Thus the displayed definitions clarify terminology but do not complete the missing hypotheses or establish a classification theorem.

Short English statement

The question asks for the geometry and order of caustics, described through optical-ray envelopes or degenerate stationary points of oscillatory phases. The supplied fragment leaves the class of examples, the meaning of order, and the intended uniform $L^\infty$ estimate unspecified, so a precise classification problem cannot yet be extracted.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001594$, title "Geometry and Order of Caustics," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposer is not identified; the supplied proposed year is 2010.
• Supplied source citation: Nonlinear Waves and Dispersive Equations (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/41, pp. 2393--2463. https://doi.org/10.4171/owr/2010/41. The collection background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no particular page or speaker is identified for this fragment.
• Additional supplied literature reference: https://www.numdam.org/article/JEDP_2010____A11_0.pdf. No author, title, or pinpoint reference is supplied for this document.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "open" and characterizes the extraction as a fragment requesting a general theory of caustics. It states that existing results classify important canonical singularities but do not turn this fragment into a single solved problem. This is supplied dated curation, not an independently verified literature assessment; the record explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming a self-contained, meaning-preserving cleanup without unsupported additions. These are supplied editorial assertions, not independent verification of the formulation or its mathematical status.