18 problems

Let $E\subset\mathbb{R}^2$ be compact, and for a direction $\theta$ let $\operatorname{Vis}_\theta E$ be the set of points $x\in E$ such that the half-line from $x$ in direction…

Assume that $C$ is a generic paracanonical curve of genus $g\geq 11$, embedded by the paracanonical bundle $\omega_C\otimes\eta$, where $\eta$ is a generic non-trivial torsion line…

Let $C\hookrightarrow\mathbb{P}^{g-1}$ be a generic canonical curve of genus $g\geq 9$, and let $p\in\mathbb{P}^{g-1}$ be a generic point. Write $C'$ for the projection of $C$ away…

Let $r\in\mathbb{N}$, let $P_j$ be the orthogonal projection onto a closed subspace $\mathscr{M}_j$ of a Hilbert space $\mathscr{H}$ for $1\leq j\leq r$, and let $P$ denote the pro…

Let $E\subset\mathbb{R}^d$ be compact. For $\theta\in\mathbb{S}^{d-1}$, let … be the closed half-line spanned by $\theta$, and define the visible part of $E$ in direction $\theta$…

David–Semmes' conjecture. For Ahlfors regular sets, the PBP and BPLG conditions are equivalent. This conjecture concerns the equivalence of a projection condition with a strong qua…

Let $K\subset\mathbb R^n$ be a convex body, with $n\geq 3$, and let $q\in\operatorname{int}K$. For each $u\in\mathbb S^{n-1}$, let $\pi_u$ denote orthogonal projection in direction…

Let $K,L\subset\mathbb R^n$, with $n\geq 3$, be convex bodies such that $L\subset\operatorname{int}K$ and $L$ is strictly convex, meaning that its boundary contains no segments. Fo…

Vertical projection dimension conjecture. For almost every $\theta \in [0,\pi)$, if $0 \leq \dim A \leq 3$, then

Katz–Tao projection conjecture. For every $\varepsilon>0$, there are $r_1,\dots,r_k\in\mathbf{Q}\cup\\{\infty\\}$, none equal to $-1$, such that

Let $K\subset\mathbb{R}^n$ be a Kakeya set, and let $G(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{R}^n$. For $\gamma\in G(n,k)$, write $\pi_\gamma$ for…

Rickard's conjecture. If all three shadows of $b3$ are cycle-free, then each shadow has at least two branch points.

Let $L^d$ be the Brownian level set and let $\mathcal{D}$ be the exceptional set defined in the source. For each direction $\theta\in[-\frac{\pi}{2},\frac{\pi}{2})$, write…

Assume that $K \subset B(0,1) \subset \mathbb{R}^{2}$ is a set with $\mathcal{H}^{1}_{\infty}(K) \sim 1$. Let $E \subset S^{1}$ be a $\delta$-separated set of directions with cardi…

Let $E\subset\mathbb{R}^3$ be a Borel or analytic set, and let $\theta:(0,1)\to S^2$ be a $C^3$ non-degenerate family of directions, meaning … for every $t\in(0,1)$. Write…

Non-degenerate-family conjecture. Any such family should satisfy the Marstrand–Mattila projection theorem.

Minimal-distance conjecture. One necessarily has

Let … be the one-dimensional Sierpiński gasket, and let $P_\theta$ denote orthogonal projection onto the line making angle $\theta$ with a fixed axis. Furstenberg's conjecture. For…