5 problems

A Gaussian stochastic field is a stochastic field whose finite-dimensional distributions are Gaussian. Its boundary behaviour means the behaviour of the field when restricted to th…

Sarnak's conjecture. There exists $\delta>0$ such that, for every $\ell\in[-\delta,\delta]$, almost surely $\{f=\ell\}\cap\mathbb{R}^3$ contains an unbounded component.

Let $f$ satisfy Assumptions and, and suppose there exist $\alpha\in(0,2)$ and $r_0>0$ such that … for all $|x|<r_0$. Let $D\subset\mathbb{R}^2$ be an open rectangle centred at the…

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 $T$ be a Gaussian eigenfunction on a compact manifold $\mathbb M$. For $u\in\mathbb R$, define the excursion set … For each Lipschitz–Killing curvature of $A_u(T,\mathbb M)$, c…