12 problems

Unstable periodic geodesics conjecture. The particular geometry of unstable periodic geodesics should play a key role in spectral-projector estimates, especially in determining opt…

Chabert's spectral-projector conjecture. The identity above should remain valid up to $\delta=\lambda^{-\frac13}$.

Pointwise Weyl-law improvement conjecture. There exists a quantifiable constant $\kappa>0$ such that, for every $x\in K_\varepsilon$,

Sharp bandwidth conjecture. The main projector estimate should hold for

Smoothed-projector expansion conjecture. Roughly, one should have

Branching-point contribution conjecture. The contribution should satisfy

Let $S$ be a polynomial surface of revolution, and let $p(t)$ be its associated polynomial. A proper real polynomial parametrization is a real polynomial map that parametrizes $S$…

Let $({\mathbb{S}}^2,g)$ be a 2-sphere of revolution with reflective symmetry, where the metric has the form … and the Gaussian curvature is considered along meridians. Gaussian-cu…

Grand uniqueness conjecture. The genus-$g=0$ or $1$ Canham problem with any isoperimetric ratio constraint $v_0\in(0,1)$ has a unique solution up to homothety. Moreover: (i) for…

Consider the Helfrich Dirichlet problem for surfaces of revolution with boundary parameter $\alpha>0$ and weight parameter $\varepsilon\geq0$. Let $\alpha_m$ denote the threshold a…

Let $\alpha>0$ be fixed, let $\varepsilon_\alpha=\frac{1}{4\alpha^2}$, and let $(u_\varepsilon)_{\varepsilon\approx\varepsilon_\alpha}$ be the smooth local family of solutions to t…

Let $N\geq 2$, and let $p$ be a trigonometric polynomial of degree $N$ with coefficients $a_N,b_N$ satisfying $a_N\ne 0$ or $b_N\ne 0$. Let $f\in\mathbb{R}[x,y,z]$ be the defining…