10 problems

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 $\gamma:[0,1]\to\mathbb S^2$ be a $C^2$ curve satisfying … For each $\theta$, let $p_{\gamma(\theta)}$ denote projection onto the line in direction $\gamma(\theta)$, and let…

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 $\theta$…

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 $K\subset\mathbb{R}^n$ 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…

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…

Full-dimensional projection conjecture. For almost every $\theta\in J$,

Let $T_m:[0,1]\to[0,1]$ be the map $x\mapsto mx\bmod 1$, and let $\Pi_{2,1}$ be the space of orthogonal projections from $\mathbb{R}^2$ to one-dimensional subspaces. Let…