59 problems

For the classical Takagi function $T:[0,1]\to\mathbb{R}$, $T(x)=\sum_{n=0}^{\infty}2^{-n}\operatorname{dist}(2^n x,\mathbb{Z})$, and the classical Weierstrass functions…

Let $\mathscr{T}$ be a hierarchical T-mesh, and let $\mathscr{C}$ be its CVR (cross-vertex relationship) graph. For a spline space with the highest order of smoothness, write…

Let $\mu$ be the self-similar measure on $\mathbb{R}$ associated to an IFS $\{\varphi_i:i\in\Lambda\}$ without exact overlaps and a probability vector $\{p_i\}$. Let $\lambda_i$ be…

A poset has a planar diagram if its cover graph can be drawn in the plane with every cover relation represented by a curve directed upwards. Its cover graph is the graph whose vert…

Gatzouras–Peres conjecture. There exists a unique ergodic $T$-invariant measure with the same Hausdorff dimension as $K$. Moreover, this measure is mixing for $T$ and is possibly m…

Let $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal I}}$ be a self-similar iterated function system on $\mathbb R$, with $|r_i|\in (0,1)$, and let $\Lambda_\Phi$ be its attractor.…

Fix $n,m\in\mathbb{N}$ and let … for $A,B\subseteq\mathbb{N}_0$. Let $r,s\in\mathbb{N}$ be multiplicatively independent, and let $A,B\subseteq\mathbb{N}_0$ be $\times r$- and…

Compactification conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ has an $m$-dimensional compactification that can be embedded into $\mathbb{R}^n$.

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

Peskine–Szpiro conjecture. Then

Let $W(\underline{0};\underline{d})$ be the locus of good determinantal schemes in $\mathbb{P}^{n+c}$ of codimension $c$ given by the maximal minors of a $t\times(t+c-1)$ matrix wh…

Lipeomorphic invariance conjecture. Metric-coordinate dimension is invariant under lipeomorphisms; that is, if $M$ and $N$ are lipeomorphic, then they have the same metric-coordina…

Real-rank inequality conjecture. For every $C^*$-algebra $A$, one has

Jacobi Bound Conjecture. The absolute dimension of $\Sigma_1$ satisfies

Let $K_t$ denote the complete graph on $t$ vertices, and let a graph exclude $K_t$ as a minor if it has no graph minor isomorphic to $K_t$. The dimension of a poset $P$, denoted…

Let $\Phi$ be an iterated function system, let $p$ be a probability vector, and let $\mu$ be the corresponding self-similar measure. The entropy rate is … where $X_1,X_2,\ldots$ ar…

Feng–Huang–Rao's conjecture. If, for some $i\in I$, $\log\alpha_i\notin\operatorname{span}_{\mathbb{Q}}\\{\log\beta_j\\}_{j\in J}$, then

Let $\partial\Delta^n$ be a finite topological space weakly homotopy equivalent to the sphere $\mathbb S^n$, with $\mathbb S^n\to\partial\Delta^n$ a trivial Serre fibration, and le…

Let $S \subset D_2^{\mathbb{N}}$ be a sofic system with primitive adjacency matrix. Let $X$ be the associated self-affine fractal, and say that $S$ has uniform complexity when the…

For integers $p,q\geq2$, let $T_p$ and $T_q$ be the corresponding transformations on $[0,1)$, and define … Assume that $p$ and $q$ are multiplicatively independent, written…

Exact overlap conjecture. If

Conjetura sobre la no contractibilidad de espacios de configuraciones. Los espacios de configuraciones $\operatorname{Conf}(X,k)$ nunca son contráctiles para $k\geq 2$.

Banaji–Fraser conjecture. Under these assumptions, $\dim$ is the Hausdorff dimension. This proposes that adding Hölder stability to the standard properties of a dimension character…

Let $f$ be a smooth diffeomorphism with a hyperbolic set $\Lambda$. For each $x\in\Lambda$, let the stable and unstable slices be the intersections of the local stable and unstable…

Equality conjecture. If $X$ is almost zero-dimensional, then