6 problems

Caustics of iterates conjecture. For $0<r\leq 1/3$, the caustic of $R_r^{2m+1}$ is a $C^\infty$ curve with exactly four cusp singularities, with two of them occurring at…

Let $C$ be an oval, meaning a smooth strictly convex closed curve in the plane, and let an $n$-th caustic by reflection from an interior point $O$ be the envelope of billiard traje…

Let $C$ be an ellipse and let $O$ be an interior point that is not a focus of $C$. For each $n\geq 1$, the $n$-th caustic by reflection from $O$ is the envelope of the family of bi…

Let $f_0=x^3+y^3+z^3$, and consider the morsification obtained from the perturbation $x^3-3\varepsilon^2x+y^3-3\varepsilon^2y+z^3$ by splitting the critical points with coordinates…

Let $f$ be a real function singularity of type $E_7$ or $E_8$, and consider the complement of the caustic in the parameter space of a versal deformation of $f$. Homology conjecture…

Let $\gamma$ be a planar billiard table that is not an ellipse. For a light source chosen inside $\gamma$, let $\Gamma_n$ be the caustic by reflection after $n$ reflections, with…