8 problems

For a function $g$ and parameters $\alpha$ and $b$, let $W_g^{\alpha,b}$ denote a prevalent $\alpha$-Weierstrass function on $\mathbb{S}^1$, and let $\operatorname{dim_H}(A)$ denot…

Let $M$ be a Riemannian manifold under Assumptions M, and let $f:M\rightarrow\mathbb{R}$ be a density function satisfying the conditions of Proposition: for some…

Let $f$ satisfy Assumptions,, and, and let $D\subset\mathbb{R}^2$ be an open rectangle centred at the origin. Write $D_R$ for its dilation by $R$, and let $N_\mathrm{ES}(D_R,\ell)$…

Let $N_\mathrm{LS}(D_R,\ell)$ denote the number of level-set components at level $\ell$ in the dilated domain $D_R$ for the Random Plane Wave. Bogomolny–Schmit variance conjecture.…

Consider the Gaussian free field $(\varphi_x)_{x\in\mathbb{Z}^d}$ on $\mathbb{Z}^d$ for $d\geq 3$, with excursion sets $E^{\geq\alpha}=\{x\in\mathbb{Z}^d:\varphi_x\geq\alpha\}$. Le…

CLE(4) level-set conjecture. The level sets of the Ginzburg–Landau model with zero boundary condition should converge to CLE(4).

Peakedness conjecture. Within the allowable range of exponents,

Abscissa generic level sets. A full Lebesgue measure set of abscissa points $x\in[0,1]$ have level sets $L(\tau(x))$ that are uncountable and have Hausdorff dimension $0$.