50 problems

Let $D$ be a bounded body in $\mathbb{R}^d$, and consider the Dirichlet problem with polynomial boundary data on $\partial D$. A polynomial is polynomial boundary data if it is the…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a strictly convex body, and let $\mathcal S$ be a family of hyperplanes that is the image of an embedding of the tangent bundle of the sphe…

Let $K\subset\mathbb{R}^n$ be a convex body with $n\geq 3$, and let $p\in\mathbb{R}^n$. Assume that every $(n-1)$-dimensional section of $K$ through $p$ is centrally symmetric, and…

Let $l$ be a nonnegative integer, and consider an ellipsoid with semiaxes described by parameters $b1$, $b2$, and $b3$. Define the sequences $N_k$, $k=0\ldots 2l$, and $\hat{N}_k$,…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)\subset\mathbb{C}^n$ be the ellipsoid defined by … For a closed Lagrangian torus $L\subset(X,\omega)$, set…

Rogers–Shephard ellipsoid conjecture. In both cases, the minimum is attained only for ellipsoids.

Let $m_0,\ldots,m_{n-1}$ be the winding numbers of a periodic billiard trajectory inside an ellipsoid, as defined by the oscillations of its elliptic coordinates. Winding-number or…

Let $\bar c_L$ be the normalized Lagrangian capacity on $2n$-dimensional ellipsoids, let $P(1/n,\dots,1/n)$ denote the cube with all $n$ radius parameters equal to $1/n$, and let…

Let $\bar c_L$ be the normalized Lagrangian capacity on four-dimensional ellipsoids, and let $B(1)$ and $Z(1/2)$ denote the ball and cylinder of the indicated radii in…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form $\omega_{\mathrm{std}}$. A Lagrangian torus…

Classification conjecture. Every closed totally magnetic submanifold of $E(\mathbf a)$ is of the form

Classification conjecture. Every closed, connected totally magnetic submanifold of $E(A)$ of positive dimension is of the form

Let $K\subset \mathbb{E}^3$ be a convex body and let $N\in K$. For a plane $\Gamma$ with $\Gamma\cap\operatorname{int}K\not=\emptyset$ and $N\notin\Gamma$, write…

Let $K,L\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $L\subseteq K$. Two-body flat-shadow conjecture. If for every $p\in\partial K$ there are at least $L(n)$ point light s…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body with boundary of class $C^3$. Local light-source conjecture. If for every $p\in\partial K$ there are at least $n+1$ point lig…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the in…

Let $K,S\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $S\subset\operatorname{int}K$. Barker–Larman conjecture. If, for every hyperplane $H$ supporting $S$, the section…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the in…

For $q\geq 1$, consider the ellipsoid … Ellipsoid existence conjecture. There is a non-planar free boundary minimal immersed disk into this ellipsoid if and only if $q>3$. This is…

Let $K$ be a centrally symmetric convex body in $\boldsymbol{R}^d$, let $F$ be a linear subspace of dimension $s\in[d]$, and let $E$ be an origin-centered ellipsoid solving the fix…

Let $S\subset \mathbb{R}^{n}$ be an embedding of $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ that is $O$-star, and let $\phi:S\to S$ map each point to the unique point on the ray from…

Let $K,S\subset \mathbb{R}^{n}$ be convex bodies, with $n\geq 3$ and $K\subset \operatorname{int} S$. For each $x\in \operatorname{bd} S$, let $\Sigma(K,x)$ denote the graze of $K$…

Let $0<a_1\leq a_2\leq\dots\leq a_n<\infty$, and let $E^{2n}(a_1,a_2,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form. Ellipsoid extremal-torus conjecture.…

Let an ellipsoid be a smooth ellipsoidal domain in $\mathbb{R}^3$, and let $\mathbb{D}_2$ denote the dihedral symmetry group acting on it. A free boundary minimal surface is a mini…

Stabilized four-ball sharpness conjecture. This embedding is sharp for all $a>\tau^4$. Proving sharpness can be reduced to an existence problem for $(p,q)$-sesquicuspidal symplecti…