48 problems

Lipschitz magnitude conjecture. Let $T\subseteq L_1$ be tractable. Then

Let $G$ be a connected graph, let $\ell(G)$ be the number of distinct graph-metric lines, and let $\textsc{br}(G)$ denote the number of bridges of $G$. A pendant edge is an edge in…

Let $(X,d)$) be a complete metric space, and let $\{f_1,f_2,\dots,f_n\}$ be a contractive family of continuous operators on $X$: for some $\lambda\in(0,1)$, every pair $x,y\in X$ h…

Stein's fixed-point conjecture. Some composition of members of $\mathcal{F}$ has a fixed point.

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…

Let $P$ be a metric space, and let $\big((M_x),\,(\mu_{xy})\big)$ be an approximate action of the pair groupoid $P\times P$, with each $M_x$ a complete metric space and $\mu_{xy}$…

Uniqueness conjecture. Every metric space $X$ with a closed and discrete set $\operatorname{dist} X$ has the property of uniqueness in the class $\operatorname{\mathcal{G\!H}}$.

Consider $d53d^n$ endowed with the metric $d(x,y)=\lVert x-y\rVert$, where the norm is any norm on $d53d^n$. Let $S_2$ be a subdivision scheme on $d53d^n$ satisfying, for…

Converse conjecture. The Lipschitz logarithmic map condition is valid only if the sectional curvature at $x^*$ is upper bounded.

Completeness. With such a topology, $\mathcal{TA}\mathcal{F}(M)$ forms a complete metric space, or is complete in a relevant sense.

Let $X$ and $Y$ be u.l.f. metric spaces, and let $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$ denote their uniform Roe algebras. An isomorphism between these algebras is said to ind…

Extension conjecture. Theorem should hold for any $s_0<D$ on arbitrary compact metric spaces.

Recall that $\aleph_{1}$ denotes the first uncountable cardinal, and let $D_{\aleph_{1}}$ be the discrete space of cardinality $\aleph_{1}$. The space…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{F}$ be the monoid of functions under composition, and let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$, and let $\mathbf{Am}$ be the monoid of…

Logarithmic competitive-ratio conjecture. The competitive ratio for metrical task systems on every $n$-point metric space is

For finite sets $A$ and $B$, define, for $p>0$, the quantity … It specializes to $J_1$ when $p=1$ and tends to $J_{\infty}$ as $p\to\infty$. Metricity conjecture. $J_p$ is a metric…

For $n\geq 3$, let $\mathcal{M}_n$ be the class of finite metric spaces whose distances lie in ${1,\ldots,n}$, and let $\mathbf{M}_n=\operatorname{Flim}(\mathcal{M}_n)$ be its Fraï…

Pointwise-convergence failure conjecture. The conclusion that $p$ is a Birkhoff-James extension fails if the assumption that $p(x_n,\,\rule{3mm}{0.15mm})$ and…

Codimension-three manifold conjecture. $X$ is homeomorphic to a manifold away from a set of codimension three.

Let $X$ and $Y$ be Banach-space-valued random variables, and let $\mathop{\mathrm{eCor}}$ denote earth mover's correlation. The full earth mover's correlation axiom conjecture. Und…

The optimal coupling conjecture. Under “general conditions”, the infimum can be attained or computed among couplings satisfying $X=X'$ or $Y=Y'$. The text presents this as a conjec…

Let $L$ be a finite set with $2leq \lvert L\rvert<\infty$, and let $G$ be a countable homogeneous complete $L$-edge-labelled graph whose automorphism group is primitive and whose a…

Let a primitive transitive homogeneous structure be given in a finite binary symmetric language, and suppose it has trivial algebraic closures. Such a structure should admit an int…

Let $(X,d_X)$ be a finite metric space with $|X|=n>1$. In the metrical task systems problem, an online algorithm receives a sequence of nonnegative service-cost functions…