29 problems

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…

Zero-measure conjecture.

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

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

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…

Dynamical-fractal universality conjecture. All fractals existing in our world are dynamical fractals.

Let $F_q(\beta)$ and $F_a(\beta)$ denote the quenched and annealed free energies of the directed polymer, respectively, let $d_s$ be the spectral dimension of the underlying fracta…

Let the underlying random walk satisfy the local limit theorem in the sense discussed by Croydon and Hambly. For $T>0$, let $Z^0_{Tn,\beta_n}$ be the directed-polymer partition fun…

Let ${\cal A}$ be a wild labyrinth pattern: its graph is connected, it has at least one vertical and one horizontal exit pair, and it satisfies the stated labyrinth-pattern connect…

Let $\{{\cal A}_k\}_{k=1}^{\infty}$ be a sequence of labyrinth patterns that are both horizontally and vertically blocked, with $m_k\ge 4$, and let $L_{\infty}$ be the resulting li…

Let $F_t$ be a bounded one-parameter affine family with attractors $A_t$, and let $t_0$ be the threshold at which attractors cease to exist. A compact set $A^*$ is an upper transit…

Infinite-length arcs conjecture. The supermixed labyrinth fractal $L_{\infty}$ generated by $\{\tilde{\mathcal A}_k\}_{k\ge 1}$ has the property that, between any two distinct poin…

Let $C$ be the category whose objects are fractals satisfying the conditions of Theorem, with suitable homeomorphisms as arrows. Let $D$ be the category whose objects are Rees mono…

Existence of self-similar embedded fractals. A fractal of the form $z \leftarrow f(z)+c$ has quasi-self-similar embedded fractals under magnification if and only if it is invariant…

Let $A\in M_2(\mathbb{Z})$ be expanding with $|\det(A)|=3$, let $f$ be its characteristic polynomial, and let ${\mathcal{D}}=\{0,1,b\}v$ with $b>1$ and $v\in\mathbb{R}^2$ such that…

Hexacarpet diffusion conjecture. The following statements hold:

Let $SG$ denote the Sierpinski gasket and $SG_{\infty}$ its infinite version, and let $M(\lambda)$ denote the spectral-mass function. Infinite-gasket asymptotics conjecture. The fu…

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…

Let $\Gamma$ be the graph under consideration, let $\mu$ be the spectral measure, and let $P_\lambda$ and $\tilde{P_\lambda}$ denote the spectral projection kernels introduced abov…