13 problems

Let $(K,d,bc)$ be a compact metric space with Hausdorff measure $bc$, and let $(\mathcal{E},\mathcal{F})$ be a strongly local regular Dirichlet form whose semigroup has heat kernel…

Topological one-dimensionality conjecture. Any set that carries a regular resistance form is a topologically one-dimensional space when equipped with the associated resistance metr…

Let a metric measure space carry a $p$-energy with $p$-walk dimension $\beta_p$, and let its associated $p$-energy measure be defined relative to the underlying measure. Singularit…

Let $p_1,p_2\in(1,\infty)$, let $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ be the corresponding Sobolev spaces, and let $U_{p_1,+}$ and $U_{p_2,+}$ be their optimal potential func…

Resistance conjecture. Nice two-sided heat kernel estimates are equivalent to the conjunction of volume doubling, $\operatorname{PI}_{2}(\Psi)$), and…

Let the Sierpiński gasket carry Brownian motion, and let the domain of its Dirichlet form consist of the functions associated with that diffusion. A function is Lipschitz when it i…

Let the Sierpiński gasket carry Brownian motion, with generator defined as the self-adjoint operator analogous to the Laplacian. A function is α-Hölder if its Hölder exponent is α.…

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

Nested-fractal variation conjectures.

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

Sierpinski carpet critical-exponent conjecture. In fact,

Assume that the fractal has three boundary vertices and that all structures have full $D3$ symmetry. Let $\rho$ be the common value of $r_jbc_j$, let $bb_3$ be the common value of…

Let $K$ be the fractal and assume the conditions of Theorems and. Let the results of Strichartz referred to in the source be the stated spectral and fractal-analytic results, norma…