83 problems

Zero-measure conjecture.

Graph fractaloid classification conjecture. The only connected finite directed graphs generating graph fractaloids are graph-isomorphic to one-vertex-multi-loop-edge graphs, one-fl…

A simple nested fractal is a fractal in the class described in the paper, equipped with its Brownian motion and associated Dirichlet form. For a fractal $F$ with Hausdorff measure…

Let $I_{k,m}$ be horizontal intervals and $J_{k,m}$ the corresponding slits in the construction of the nowhere dense set $E\subset\mathbb R$. The Green function of…

Kigami–Lapidus lattice-case conjecture. The limit

Let a p.c.f. self-similar set mean a post-critically finite self-similar set, and let a finitely ramified self-similar set be a self-similar set with finitely ramified cell structu…

Equivalence conjecture. For all finitely ramified fractals, the condition

Let $F$ be a finitely ramified fractal considered in the paper, let $\nu$ be its energy measure, and let $Z(x)$ be the matrix defining the weak gradient metric. The generalized Rie…

Let a spectral zeta function of a symmetric fractal have a product structure with a “polynomial” part and a “geometric” part, and suppose that poles of the polynomial part may be c…

Let $M_Fcsubset\mathbb T^3$ be a connected smooth two-manifold homologous to zero, and let a magnetic field determine trajectories on it. For a smooth one-parameter family…

Fractal-window conjecture. All square-triangle tilings with 12-fold symmetry obtained by projection require a fractally shaped window. In a certain sense, the window in the example…

The two-dimensional Sierpinski carpet is a fractal space for which the paper's resistance-form and heat-kernel framework may require modifications. Applicability conjecture. The re…

Palis's conjecture. If

No-interior conjecture. For every $p\geq 2$ and every $a,b\in\mathbb{R}$,

Let $K$ be a p.c.f. self-similar set with spectral dimension $d_S$, and let $N(u)$ denote the number of local extrema of an eigenfunction $u$ with eigenvalue $\lambda$. Spectral-di…

Let $\check{\mathbb{F}}_c$ be the class of spaces under consideration and let the stochastic processes associated with its elements be equipped with the identity referred to as the…

Let $K$ be a knot, and let the Sierpinski Tetrahedron denote the fractal obtained by the iterative construction described in the paper. A finite iteration is one of the finite-stag…

Equality conjecture. If $A$ is a strictly self-similar fractal with dimension $d$, then

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 $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$, and suppose $\mathcal{L}f$ is Hölder continuous on $\mathrm{SG}$. Let $\mathcal{L}_\epsilon$ and…

Let $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$. Consider the rescaled averaging Laplacian and the rescaled graph Laplacian at random sample points, both de…

Let $\mathcal{L}$ be the Laplacian on the Sierpiński gasket, whose relevant Neumann eigenspace has multiplicity two, and consider eigenmaps arising from random graph approximations…

Let $\mathrm{SG}$ be the Sierpiński gasket, let $\mathcal{L}$ be its Laplacian, and let $\mathcal{L}_\epsilon(\omega)$ be the random graph averaging Laplacian defined from random s…

Let $\mathrm{SG}$ be the Sierpiński gasket with its standard Laplacian $\mathcal{L}$, and define the cellular semimetric by … Let $\mathcal{L}_\epsilon$ be the averaging operator o…

Let $\Phi$ be a self-similar iterated function system on $\mathbb R^2$ and let $K$ be its attractor. The Luo–Rao–Xiong local connectedness conjecture. If $\Phi$ satisfies the open…