89 problems

Let $s$ and $k$ be positive integers, let $\varepsilon_i\in\{\pm1\}$ for $1\leq i\leq s-1$, and consider the closure of the braid … A generalized rosette knot is this closed braid.…

Billiard-flow deviation conjecture. For all rational-angle Euclidean polygons, the deviation of ergodic averages for the billiard flow is the same for almost all directions, and de…

Feingold–Peres off-diagonal variance conjecture. For ergodic flow, as $E\to\infty$,

Let $\phi_n$ be Laplacian eigenfunctions on an ergodic billiard, and let a quantum limit mean a measure to which $|\phi_n|^2$ converges weakly. Quantum unique ergodicity conjecture…

Let $\Omega$ be a billiard domain and let $\hat{A}$ be multiplication by a test function $A$. Define the local diagonal variance $V_A(E)$ as the mean square of…

Let $\Omega$ be the billiard domain, let $\hat{A}$ be multiplication by a test function $A$, and define its spatial average by … For eigenfunctions $\phi_n$ of the Laplacian with e…

Let $\gamma\subset\mathbb{R}^2$ be an ellipse, let $O$ be a light source inside $\gamma$ and different from a focus, and for each $n\geq 1$ let $\Gamma_n$ be the envelope of rays f…

Let $B$ be the unit disk in the plane, and for a convex shape $K\subset\mathbb{R}^2$ let $c(K\times B)$ be the length of the shortest closed generalized billiard orbit in $K$. Mall…

Let $\rho_0$ and $\rho_1$ be two distinct rotation numbers in $(0,1/2]$. For ellipses $\mathscr E$ and $\mathscr E'$, let $\beta_{\mathscr E}(\rho)$ denote the Mather beta function…

Invariance conjecture. For every $R_0\in\mathbb{R}^d$, $b\in S^{d-1}$ and $p\in(0,1)$, the rescaled trajectories $R_t^{(n)}$, $t\in[0,1]$, converge to isotropic Brownian motion in…

Let $p\in(0,1)$ be fixed, let $b\in S^{d-1}$ be a direction, and let $\sigma_{b,p}$ denote the standard deviation associated with the random billiard walk. A direction is called ra…

Let $b\in S^{d-1}$ be a direction, let $p\in(0,1)$ be the parameter, and let $\sigma_{b,p}^2$ denote the variance of the random billiard walk in direction $b$. Monotonicity conject…

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with smooth boundary, and let $p=1,2,\dots$. A billiard trajectory is a trajectory in $\Omega$ satisfying the billiard reflecti…

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain whose boundary $\partial\Omega$ is a $C^2$-smooth convex curve. Suppose that a neighborhood of $\partial\Omega$ is foliat…

Let $\Omega\subset\mathbb{R}^2$ be a chaotic billiard, so its billiard flow is ergodic and has positive topological entropy. Let $\{u_n\}$ be an orthonormal sequence of eigenfuncti…

Let a convex polygon billiard be a billiard in a convex polygonal table, and let $W_t(P)$ denote the billiard wave front at time $t$ from a point $P$ in the table. The wave front b…

Let a convex billiard table be a convex planar domain, and let $W_t(P)$ denote the billiard wave front at time $t$ from a point $P$ in the table. The wave front becomes dense if it…

Let $\mathcal{A}$ denote the attractor describing the generic behaviour near the initial singularity, let $\mathcal{K}$ be the Bianchi type I Kasner subset, and let…

Let $\Omega$ be a strictly convex domain with smooth boundary, and let its billiard map act on the cylinder parametrized by…

Let $\Omega\subset\mathbb{R}^2$ be a plane region with smooth boundary, and consider the billiard in $\Omega$. The periodic points are the points of the billiard phase space corres…

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 $\Omega \subset \mathbb{R}^2$ be a smooth, bounded, strictly convex billiard table, and let $\Gamma \subset \Omega$ be a convex caustic of length $|\Gamma|$ with Lazutkin param…

Let $D$ be the domain under consideration, and let $\triangle P_1P_2P_3$ be a triangle in $D$. Write $\psi_{\triangle P_1P_2P_3}$ for its associated billiard map,…

Smoothness conjecture. The solution $u(t)$ of the Poincaré problem is smooth.