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...

Is there an effective algorithmic procedure to produce and recognize a hyperbolic knot of depth $n$ for any given $n$? What about $\ge n$?...

Given a collection $\mathscr{C}$ of topological or geometric types of surface, what $3$–manifolds admit a taut foliation $\mathscr{F}$ whose leaves ar...

Let $X$ be a vector field on a $3$–manifold. When is there a foliation $\mathscr{F}$ of $M$ transverse to $X$?...

Generalize the Teichmüller polynomial from the fibered faces of the Thurston norm ball to the other faces (of some possibly generalized polyhedron, pe...

Given a collection of taut foliations $\mathscr{F}_i$ of $M$, what are the obstructions to finding a metric on $M$ for which the $\mathscr{F}_i$ (afte...

How many Reeb components must a foliation of an open $3$–manifold contain?...

What generalizations of the notion of taut foliation make sense on an open $3$–manifold?...

Characterize those essential laminations which contain genuine sublaminations....

Suppose $\Lambda$ is a full genuine lamination; i.e. it has some complementary region which is an ideal polygon bundle over a circle. Suppose $M$ is h...

Suppose $\Lambda$ is a genuine lamination. When can $\Lambda$ be ``filled in'' to a very full lamination $\Lambda'$? Does it help for $M$ to be hyperb...

Are loosesse laminations good for anything? Are leaves of the universal cover of a loosesse lamination properly embedded? If $M$ contains a loosesse l...

Give an example of a lamination in an atoroidal manifold –- perhaps loosesse –- which can never be realized by minimal surfaces for any metric, but wh...

Characterize branched surfaces embedded in $3$–manifolds which can be non–trivially split to a homeomorphic copy of themselves....

Develop a theory of hierarchies for branched surfaces....

When does a Haken sum operation make sense for a pair of laminations in normal form with respect to a fixed triangulation?...

Let $M$ be a $3$–manifold, and $\Lambda$ an essential lamination. Let $C$ be a cycle representing the fundamental class of $M$. Is there a cycle $C'$ ...

Suppose $\mathscr{B}$ is a branched surface in $M$ which is dual to a taut local orientation. Is there a finite cover of $M$ in which the pullback of ...

Give a useful definition of thin position for an embedded graph $\Gamma \subset M$ with respect to a taut foliation $\mathscr{F}$. If $\Gamma$ is the ...

Suppose $\mathscr{F}$ is an $\mathbb{R}$–covered foliation of an atoroidal $3$–manifold $M$. Is the holonomy representation $\rho_H$ of $\pi_1(M)$ on ...

For a fixed manifold $M$, describe the structure of the set of all essential laminations with a transverse $\widetilde{SL(2,\mathbb{R})}$ structure....

Suppose $M$ admits a minimal taut foliation. What is the best analytic (transverse) quality of a taut foliation it admits? Can we find a minimal folia...

Is there a universal constant $c$ such that a hyperbolic $3$–manifold $M$ whose fundamental group $\pi_1(M)$ can be ordered out to radius $c$ can be l...

Let $\mathsf{T}$ be some class of abstract computers; e.g. finite state automata, Turing machines, Turing machines relative to some oracle $O$, etc. A...

Let $\Lambda^\pm$ be a pair of laminations of $S^1$ which are transverse to each other and have finite area complementary domains. Suppose $\Gamma$ is...

What is the best analytic quality for the action of $\pi_1(M)$ on a universal circle $S^1_\mathrm{univ}$?...

Is there a universal transverse surgery description of tautly foliated manifolds, in the sense that there is a fixed $M$ such that for every tautly fo...