15 problems

Let $f$ be a complex rational function, and let $J(f)$ denote its Julia set. A compact metrizable space is finitely self-similar if it arises from a finite self-similarity system,…

A finite simplicial complex is the topological space obtained by gluing finitely many simplices along faces. A compact metrizable space is finitely self-similar if it arises from a…

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

Arias's conjecture. The set $A$ has finite symmetric difference with $E$, and the one-sided difference $A\setminus E$ is a finite subset of

Self-similarity conjecture. For all primes $p\geqslant 5$, no open subgroup of $SL_1^1(\Delta_p)$ is self-similar. For all primes $p\geqslant 3$, no open subgroup of…

Let $A=\operatorname{Alg}(v_0,v_1)$ be the associative superalgebra corresponding to the Lie superalgebra $\mathbf R=\operatorname{Lie}(v_0,v_1)$, where $v_0,v_1$ are defined by th…

Let $\mathbf R=\operatorname{Lie}(v_0,v_1)$ be the Lie superalgebra whose generators $v_0,v_1$ are defined by the pivot construction referenced in the source. A Lie superalgebra…

Let $P$ be the solution of the ODE defining the proposed blow-up profile, and let $u(x,t)$ solve the generalized Korteweg–de Vries equation for $n=5$ with smooth initial data…

Quasi-fractal intersection conjecture. The Zariski closure of $Z(\bar K)\cap Q$ is a union of finitely many points and finitely many components $B_i$ such that, for each $i$, the i…

Arithmetic-fractal intersection conjecture. For each such component $B_i$, either $B_i$ is a point, or $B_i(\bar K)\cap F$ is an arithmetic fractal with respect to some induced end…

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…

Fractional Brownian motion self-similarity conjecture. The tree $\operatorname{shape}(\operatorname{tree}(B^H))$ is Tokunaga self-similar with

Let $\psi$ be a peak-type singular solution of the supercritical biharmonic nonlinear Schrödinger equation. Let $S_{\mathrm B}(\rho)$ be a radial self-similar profile satisfying ……

Let the full butterfly be the parameterized spectrum of the almost Mathieu operators, including irrational values of the vertical parameter $\theta$, and let $S=S_{M,r,\pm}$ be a s…

Shrinking-ring conjecture. The solution is quasi self-similar, $\psi\sim\psi_{Q_{\mathrm B}}$, for $r-r_{\rm \max}=\mathcal{O}(L)$; it is a shrinking ring, with…