Unsolved Problems

Determine the algebraic structure of the concordance group $\mathcal{O}$ of open strings. (a) Is it abelian? (b) Does it contain torsion?...

For a classical knot, does its slice genus as a virtual knot agree with its slice genus as a classical knot?...

Let $K$ be a hyperbolic knot in $S^{3}$ and $\chi(K)$ the space of con- jugacy classes of $\operatorname{PSL}_{2}(\mathbb{C})$ representations of $\pi...

(a) Every connected cubic $($ i.e. trivalent $)$ graph has freeness index at least 2. (b) Every graph has freeness index at least two. (c) There is a ...

Is every fibered link in $S^{3}$ realized as the link of an isolated singular point of a polynomial map $\mathbb{R}^{4} \to \mathbb{R}^{2}$?...

Are there infinitely many congruence arithmetic links in the 3-sphere?...

(a) Fix a long link L. What is the homotopy type of the embedding space of links isotopic to L? (b) Fix a link L in a 3-manifold M. What is the homoto...

(Ivanov conjecture). Let $S$ be an orientable surface of finite type with genus at least three. If $G \leq \operatorname{Mod}(S)$ is a subgroup of fin...

(Congruence subgroup problem). Does every finite-index sub- group of the mapping class group of $S$ contain a congruence subgroup?...

Is the mapping class group of a surface of finite type linear?...

Let $S_{1}$ and $S_{2}$ be orientable surfaces of finite type. Under what conditions do injective maps from (finite-index subgroups of) the mapping cl...

For $g \geq 3$, determine a finite presentation for the Torelli group $\mathcal{I}_{g}$, or show that no finite presentation exists....

Give a classification or enumeration of the finite-index sub- groups of $\operatorname{Mod}(S_{g})$ that are generated by Dehn twists, Dehn multitwist...

Classify the homomorphisms from the braid group $B_{n}$ on $n$ strands to the braid group $B_{m}$ on $m$ strands, where $n, m \in \mathbb{N}$ are arbi...

Fix distinct trivial tangles $\tau_{1}, \tau_{2}$ for which $\tau_{1} \cup\tau_{2}$ is the unknot. Describe the intersection of the associated wicket ...

Is there a nice presentation of the $n$-stranded braid group whose generating set is the set of all positive elementary braid half-twists?...

(a) Is there an efficient algorithm to compute distances in the curve complex of a surface? The input to the algorithm should be the surface and the c...

Find precise estimates for both the extremal and average behav- ior of the simple lifting degree of curves, in terms of combinatorial (e.g., intersect...

(a) What is the maximum number of systoles on a closed, hyperbolic surface of genus $g$? (b) What is the maximum cardinality of a set of pairwise non-...

Let $S$ be a surface, and let $\Gamma_{1}, \Gamma_{2}$ be isotopy classes of embedded graphs in $S$. Determine when $\Gamma_{1}$ and $\Gamma_{2}$ are ...

(a) Does every Jordan curve in the Euclidean plane contain the vertices of a square? (b) Does every Jordan curve in the Euclidean plane contain the ve...

Is the genus $g$ Goeritz group $\mathcal{G}_{g}$ finitely generated when $g \geq 4$? If so, find a set of generators....

If two hyperbolic surfaces have the same unmarked simple length spectra (i.e., the same multiset of lengths that correspond to simple closed curves), ...

Suppose that $c$ is a geodesic current on a hyperbolic surface, and suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same ...

What is the best lower bound on the volume of a fibered hyper- bolic 3-manifold one can give in terms of the translation length of the monodromy with ...

Which mapping classes give rise to arithmetic hyperbolic 3- manifolds as their mapping tori?...

Can one detect holomorphicity from a monodromy factoriza- tion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are the spec...

What is the minimum number, $m_{g,b}$, of right-handed Dehn twists along essential curves into which the boundary multi-twist, $\Delta:= T_{\delta_{1}...

Does every Lefschetz fibration over the 2–sphere admit a sec- tion?...

Does there exist a surface bundle over a surface that admits a complete hyperbolic metric, or, more generally, a complete metric of variable negative ...

Does there exist a complex surface $X$ that admits three or more non-isomorphic structures as a surface bundle over a surface?...

Consider surface bundles over surfaces where both fiber $F$ and base $B$ have genus $\geq 2$ and where $\pi_{1}(B)$ injects in the mapping class group...

(Kontsevich–Zorich conjecture). Understand the homotopy types of strata of abelian differentials. Which stratum-components are $K(\pi, 1)$ spaces? Wha...

For $n \geq 4$, does the braid group $B_{n}$ admit a finite-index sub- group that embeds in a right-angled Artin group?...

Let $\Gamma$ be a graph that is not a nontrivial join, and let $A(\Gamma)$ be the associated right-angled Artin group. Does there exist an injective m...

Determine the Artin groups that can be embedded into a map- ping class group....

Which right-angled Artin groups contain closed hyperbolic sur- face groups? Is there an algorithmic or graph-theoretic criterion to decide this?...

Let $S$ be a closed surface of genus at least 2. Show that the stable commutator length is rational on the commutator subgroup of $\pi_{1}(S)$....

Does every surface bundle over a surface admit a flat connec- tion? What about surface bundles over 3-manifolds?...

Let $S$ be a closed compact surface (without boundary). Is there a finitely generated, torsion-free group $G$ such that $G$ cannot act faithfully by h...

Let $S$ be a compact surface. For $0 \leq r < s$, does there exist a nontrivial finitely generated subgroup $G_{r} \leq \operatorname{Diff}^{r}_{0}(S)...

Is the first-order theory of the mapping class group of a surface decidable?...

Are systems of equations over mapping class groups and braid groups decidable?...

Give a Nielsen–Thurston-type classification for the mapping class groups of infinite-type surfaces. In particular, which homeomorphisms are the approp...

Give an appropriate analogue of the curve graph for infinite- type surfaces, and characterize the surfaces for which no such graph exists....

(a) Does the mapping class group of an infinite-genus surface with no planar ends contain every countable group? (b) Does the mapping class group of t...

(a) Let $S$ be an infinite-type surface and $\varphi$ a mapping class for which there is a (marked) conformal structure $\Sigma$ on $S$ with respect t...

Give a finite list of practically computable invariants of the mapping class group or pure mapping class group of an infinite-type surface $S$ that de...

Is the geodesic flow in almost every direction on the Chamanara surface ergodic? What about on the translation surface considered by Bruin and Lukina,...

Let $X$ be a compact, totally disconnected subset of $\mathbb{R}^{2}$ with $|X| \geq 2$, and let $\Gamma_{X}$ denote the mapping class group of $\math...