5 problems

Critical nodal-set conjecture. The set $\{f=0\}$ has no unbounded connected components.

Let $G$ be a planar graph, and consider i.i.d. Bernoulli site percolation at parameter $p=\tfrac{1}{2}$. Benjamini–Schramm Conjecture 8. If percolation occurs almost surely, meanin…

Let $Q$ be a quad and let $\alpha$ be a smooth curve. For a discrete percolation measure $\mu_\eta$, let $N_\eta=N_\eta(Q,\alpha)$ be the minimal possible number of intersections w…

Let $Q$ be a quad and let $\alpha$ be a smooth curve. A crossing of $Q$ is a percolation crossing between the designated boundary arcs of the quad. Finite intersection property. In…

Super-strong separation lemma. Conditioned on at least four of these interfaces reaching $B(R_1)$, the set of interfaces reaching $R_1$ has positive quality with positive probabili…