7 problems

Let $L$ be the curvature length scale and $\ell_0$ the lattice edge length for regular or semi-regular hyperbolic tessellations, and let $n$ be the polygon length. Hyperbolic-latti…

Let $\nu$ and $\vartheta$ be the critical exponents governing the continuum limit and the zero-cosmological-constant partition function of the SOP model. SOP critical-exponent conj…

Let a self-overlapping polygon have boundary length $2n$, and let its multiplicity be the number of distinct distorted disks it bounds. Exponential multiplicity conjecture. There e…

Let $w_{2n}$ be the number of self-overlapping polygons of length $2n$ in the SOP model, counted with multiplicity factor $\mu_\gamma$. SOP connective-constant conjecture. The limi…

In dimension $3$, let $K^\text{Arm}(n,w,k,r)$ be the probability that a length-$k$ arc of a random open arm is an $r$-local knot, and let $K^\text{Pol}(n,w,k,r)$ be the correspondi…

Let $\operatorname{Arm}(n,d,w)$ be the space of open $n$-edge arms in $\mathbb{R}^d$ with fixed edgelength $w$, and let $\operatorname{Pol}(n,d,w)$ be the corresponding space of cl…

Let $L_n$ be the random polygon sequence generated by the consecutive random subdivision, and suppose that $u$ satisfies … and that the support of $\xi$ contains at least two dist…