36 problems

Bell–Ho–Strichartz conjecture. As $m\to\infty$, the law of $r\circ c^{(w)}$ under $\lambda_m$ converges to the Dirac measure at $\sqrt{8/75}$, while the law of…

Nonvanishing normal derivative conjecture. For every integer $t\geq 0$,

Asymptotic growth constant conjecture. The asymptotic growth constant is

Spanning-tree factorization conjecture. The number of spanning trees is

Let $K$ be the Sierpiński gasket. For a nonconstant harmonic function $h$ on $K$, let $uh=muh/muh(K)$ be its normalized energy measure. For nonconstant harmonic functions $h1,h2$…

Let $SG$ be the Sierpiński gasket, let $L$ be its bottom line, and let ${\Greekmath 011A}_D^{SG\setminus L}(x)$ denote the Dirichlet eigenvalue counting function on $SG\setminus L$…

Let the nonlinear Schrödinger equation on the Sierpinski gasket be the equation considered in the paper, and let its flow map assign to initial data the corresponding solution. Non…

Strichartz's extension conjecture. The corresponding existence and uniqueness theorem extends from the Sierpinski gasket to harmonic maps from p.c.f. sets to the circle.

Let $K$ be the Sierpiński gasket, let $d$ be its metric, let $d_h$ and $d_w$ denote its Hausdorff and walk dimensions, respectively, and let $G_s$ be the Riesz kernel associated wi…

Numerical conjecture. The spherical Hausdorff measure satisfies

Lipschitz-function conjecture. Every non-constant Lipschitz function is not of bounded variation with respect to definition $(A)$.

Let $G_\infty$ be the infinite Sierpinski gasket lattice, let $\mathcal{U}$ be the spectral-decimation map, and let $\mathcal{J}(\mathcal{U})$ denote its Julia set. For parameters…

Let $\mathsf{SG}$ be the Sierpinski gasket graph, let $\mathcal{B}_n$ be the graph-metric ball of radius $n$, and let $\mathcal{I}_{|\mathcal{B}_n|}$ be the internal DLA cluster fo…

Boundary-value decay conjecture. For $r\neq 1$ and $r\neq\frac{\sqrt{17}-3}{4}$,

The ratio-existence conjecture. For all but at most finitely many values of $r$,

The asymptotic ratio conjecture. For $r\neq \frac{\sqrt{17}-3}{4}$,

The asymptotic ratio conjecture. For $r\neq 1$ and $r\neq \frac{\sqrt{17}-3}{4}$,

Entropy comparison conjecture. The Tower of Hanoi entropy is strictly smaller than the Sierpinski gasket entropy:

Let $B_o(r)$ be a nonempty initial subset of the Sierpinski gasket, and independently place either $4-\delta$ or $4+\delta$ chips at each vertex with probability $1/2$. The resulti…

Fix $k\in[4,12)$, let $K$ be the limit fractal of the Sierpinski gasket, and let $\nu$ be its standard self-similar measure. Consider the rescaled sandpile configurations … These c…

Let $SG$ be the graphical Sierpinski gasket, let $o$ be its source vertex, and let $B_o(n)$ denote the graph-metric ball of radius $n$ centered at $o$. Write…

Local-extrema count conjecture. The numbers of local extrema of the restrictions of $\psi_n^{(2)}$, $\psi_n^{(5)}$, and $\psi_n^{(6)}$ to the bottom edge of $\mathcal{SG}$ are, res…

Let $K_H$ be the harmonic Sierpinski gasket equipped with its geodesic metric, and let the direct sum spectral triple be the one constructed from the constituent curves. The associ…

Magnetic energy convergence conjecture. The convergence

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…