14 problems

Convex-cocompactness conjecture. Then $G$ acts discretely, cocompactly, and isometrically on a convex subset of $\mathbb H^3$ with nonempty totally geodesic boundary.

Let $A_\infty$ denote the final retained set in planar fractal percolation, and let $p_c$ be its critical parameter. Critical fractal-percolation carpet conjecture. At the critical…

Let $(K,d,m)$ be the Sierpiński carpet, let $d_{\mathrm{ARC}}(K,d)$ be its Ahlfors regular conformal dimension, and suppose this dimension is attained. Let $\mathcal{V}_0$ be the b…

Let $(K,d,m)$ be the Sierpiński carpet, and consider the axiomatic characterization of self-similar $p$-energies in Proposition 4.3. Assumption (g) there is the unit contractivity…

Let $(K,d,m)$ be the Sierpiński carpet, let $p=d_{\mathrm{ARC}}(K,d)$ be an attained Ahlfors regular conformal dimension, and let $\Gamma_p\langle f\rangle$ denote the $p$-energy m…

Let $(K,d,m)$ be the planar standard Sierpiński carpet, and let $p\in(1,\infty)$. A self-similar $p$-energy is a pair $(\mathscr{E}_p,\mathscr{F}_p)$ satisfying the paper's self-si…

Generalized Sierpinski carpet criticality conjecture. For generalized Sierpinski carpets and similar fractals,

Sierpinski carpet weak Bakry–Émery conjecture. The Sierpinski carpet satisfies $wBE(\kappa)$ for some

Sierpinski carpet critical-exponent conjecture. In fact,

Nonexistence conjecture. Unlike in the homogeneous case, a closed formula cannot be obtained for the optimal sets of $n$-means for the nonhomogeneous probability distribution consi…

Consider the three compact fractafolds obtained from the Sierpinski carpet by boundary identifications corresponding to the torus, Klein bottle, and projective plane, and let their…

Let $N_1(t)$ and $N_2(t)$ denote the eigenvalue counting functions of two of the compact fractafolds obtained from the Sierpinski carpet by boundary identifications corresponding t…

Let $F$ be the two-dimensional standard Sierpinski carpet, realized by two self-similar structures that yield the same Laplacian. Let $d_f$ be its fractal dimension and $d_w$ its w…

Let a generalized Sierpinski carpet have self-similar, possibly graph-directed, structures yielding the same Laplacian operator. Let $d_k$ denote the relevant dimensions, with…