8 problems

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…

The side-length extrema conjecture. There are real values $1<a_1^-<a_1<a_1^+<a_2^-<a_2<a_2^+<\sqrt{2}$, with $a_1\approx1.3299$ and $a_2\approx1.4123$, such that…

Let $\mathcal E_h=\{(x,y):(hx)^2+y^2<1\}$ be an ellipse with $h\in(0,1]$, and consider its Dirichlet or Neumann Laplace operator. Hezari–Zelditch's conjecture. The eigenvalues of a…

Let a Poncelet triangle family be inscribed in a circle with inconic or caustic, and let $F$ vary over the circumcircle. For each $F$, let $W$ be the envelope point of the correspo…

Let a triangle center be defined by barycentric coordinates that are rational functions of the squares of the triangle's sidelengths, and consider its locus over the relevant famil…

Consider the family of ellipses … For each such ellipse, consider the multiplicities of the eigenvalues of its Laplacian. Generic multiplicity-two conjecture. For a generic class o…

Area inequality conjecture. For any such ellipse and quadrilateral,

Let $R$ be a convex quadrilateral, and consider ellipses circumscribed about $R$. Minimal-eccentricity circumscribed ellipse conjecture. The center of the ellipse of minimal eccent…