16 problems

Analyticity conjecture. The map $t\mapsto k(t)$ is analytic everywhere except for at most $d-1$ points.

Let $\pone$ denote the projective line, let $F$ be a function on $\pone$, and let $\omega$ be a two-form on $\pone$. Assume that $F$ and $\omega$ are analytic in the real-analytic…

Analyticity conjecture. The map $t\mapsto k(t)$ is analytic everywhere except possibly at $t_0$.

Let $u$ be a solution of the Zakharov–Kuznetsov equation, and let the moving region be the region described above in which the solution exhibits almost instantaneous smoothing unde…

For a direction $s$, let $\lambda_{\mathrm{sat}}(s)$ be the saturation threshold and let $\lambda_{\mathrm{exp}}$ be the threshold for quasi-long-range order. Let $\nu_s(\lambda)$…

Analyticity conjecture. $S_{(((31)6)(24))(57)}(a_1,a_2,a_3,a_4,a_5,a_6,a_7)$ is the expansion of a real analytic function $\phi(z_1,\dots,z_7)$ on $X_7$ in this domain.

Let $\mathcal{E}^\alpha$ denote O'Hara's knot energy for $2<\alpha<3$. The preceding smoothness result states that finite-energy critical points are smooth, and analyticity is know…

An irrational vector $\rho\in\mathbb{R}^d$ is Diophantine of class $\beta$ if there exists $C_\beta>0$ such that, for every $k\in\mathbb{Z}^d$ with $k\neq 0$ and every…

Let $u_*$ be the solution of the perturbation equation on $(0,S)\times\overline{H}$, where $H$ is the spatial half-space, arising from sufficiently small Lipschitz initial data. Th…

Let $\rho$ be a distributional solution of the porous medium equation whose pressure $\rho^{m-1}$ is smooth up to the relevant parabolic boundary and whose level sets are analytic.…

Consider the formal time-Taylor series for the Lagrangian fluid particle trajectories arising from Pohle's Lagrangian formulation of the incompressible Euler equations. Stoker's co…

Let $\operatorname{AdS}_3$ denote three-dimensional Anti-de Sitter space, and consider small smooth perturbations whose spatial dependence admits a radius of analyticity $\rho(t)$,…

Let the initial datum be analytic in $x$ and $C^2$ in $Y$, and consider the corresponding Prandtl solution, which remains regular for a short time under these hypotheses. Analytici…

The propagation of analyticity concerns solutions of a weakly hyperbolic nonlinear system … Assume that the solution $U(t,\cdot)$ is bounded in a functional class $\mathcal X$ in w…

Let $r>3/2$, let $u0 otin H^r$?...

Let $u0 e?...$