Mathematics Problem Archive

Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property? A set has the fixed poi...

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group....

Conjecture $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \t...

Conjecture (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensu...

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal...

Conjecture The poset of completary multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See belo...

Conjecture The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See below for defin...

Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right): \mathfrak{F} \left( U_j \right)...

Conjecture Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\operatorname{Src}f_1=\operatorname{Src}f_2=A$. Then there exists a poi...

Conjecture Let $f\in Diff^{r}(M)$ and $p\in\omega_{f}$. Then for any neighborhood $V_{f}\subset Diff^{r}(M)$ there is $g\in V_{f}$ such that $p$ is pe...

Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected, compact and boundaryles manifold M and $\chi^{r}(M)$ the space ...

Conjecture For composable reloids $f$ and $g$ it holds - $\operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f$ if $f$ is a co-compl...

Conjecture Every metamonovalued funcoid is monovalued. The reverse is almost trivial: Every monovalued funcoid is metamonovalued....

Conjecture Every metamonovalued reloid is monovalued....

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \}: \left( X \cup Y ...

Conjecture Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric,...

Definition $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f; \operatorname{Dst} f)} f$ for reloid $f$. Conjecture $(\mat...

Let $\mathfrak{A}$ be an indexed family of sets. Products are $\prod A$ for $A \in \prod \mathfrak{A}$. Hyperfuncoids are filters $\mathfrak{F} \Gam...

Conjecture $f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sou...

Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\in...

Conjecture For every composable funcoids $f$ and $g$ $$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{R...

Define for posets with order $\sqsubseteq$: - $\Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; ...

Warning: This formulation is vague (not exact). Question Characterize the set $\{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_...

Problem Let $\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal? 1. The funcoid corresponding t...

Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum ...

Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?...

Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic....

Does there exist a totally skew embedded $3$-disc in $\mathbb{R}^7$?...

(Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?...

(Gromov) Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?...

(Gromov) For a given n is there an example of a hyperbolic group of dimension n in which every infinite index subgroup is free? Or in which there are ...

(Swarup) Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ and has finite index in its normalizer. Assume that there is $n>0...

(Whyte) Let Γ be a 1-ended hyperbolic group which is not virtually a surface group. Can every infinite index subgroup be free?...

Is every word-hyperbolic group residually finite?...

(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of $G$, and let $\operatorname{rank}(G^n)$ be its smallest number of generators. If $...

(Dani Wise) Find 'nice' groups, for example CAT(0) or automatic groups, for which $\operatorname{rank}(G^n)$ does not tend to infinity as $n\to\infty$...

(Ian Leary) Is there a version of the Kan-Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any fin...

(Gromov) If $G$ admits a finite-dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?...

(Eilenberg-Ganea) Is there a group G of cohomological dimension 2 and geometric dimension 3?...

Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y . Croke-Kleiner have examples where the bou...

Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?...

Does every Artin group have a finite $K(G,1)$?...

(Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting properly discontinuously by isometries on $X$. (1) Ca...

(Ruth Charney) Classify Coxeter groups up to isomorphism....

(Ruth Charney) Classify Artin groups up to isomorphism....

(Ruth Charney) Are all [finite type] Artin groups CAT(0)?...

(Ruth Charney) Are all Artin groups automatic?...

(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?...

Characterize 1-relator groups which are limit groups....

(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said abou...