51 problems

Kronheimer–Mrowka conjecture. Appropriate versions of instanton Floer homology and Heegaard Floer homology are isomorphic.

Let $S=S_{g,r}^n$ be an orientable surface of genus $g\ge 3$ with $r$ punctures and $n$ boundary components, and let…

Let $K\subset S^3$ be a knot. It is slice if it bounds a smooth properly embedded disk in the standard $4$-ball $B^4$, and it is ribbon if it bounds a disk that can be isotoped so…

Let $S$ be a rational homology projective plane with quotient singularities, and write … The smooth locus $S^0$ is assumed to be simply connected. Kollár's algebraic Montgomery–Yan…

Witten's finiteness conjecture. The vector space $Sk(M)$ is finite dimensional over $\mathbb{Q}(A)$.

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Waldhausen's virtual Haken conjecture. $M$ is finitely covered by a Haken manifold. This…

Rasmussen's rank conjecture. The rank of reduced Khovanov homology is always greater than or equal to the rank of knot Floer homology:

Let $K$) be a knot, let $D(K)$ denote its positively-clasped Whitehead double, and let $[K]$ and $[D(K)]$ denote their elements in the knot concordance group $\mathcal{C}$. Kirby’s…

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Virtual positive Betti number conjecture. Some finite cover of $M$ has positive first Bet…

Let $Y$ be a closed oriented 3-manifold satisfying … If $Y$ is not $S^3$, then Kirby's $SU(2)$ representation conjecture. there exists a homomorphism … with non-abelian image. This…

Let two spheres in $\mathbb{R}^3$ be equipped with oriented normal bundles and intersect along a circle in the space $\tilde Y$, with the canonical orientation induced on the inter…

Let ${\mathcal X}_{\kappa}$ be the character variety with parameter $\kappa$, and let ${{\mathcal E}(\rho)}$ be the set of end invariants of a character $\rho$. The ending invarian…

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character and let ${{\mathcal E}(\rho)}$ denote its set of end invariants, viewed as a subset of the projective lamination space…

Neumann's surjectivity conjecture. If $k=1$, then $\mathbb{U}$ is onto. In general, every $\mathsf{PSL}(2,\mathbb{R})$-representation with dense image lies in…

Rudolph's characterization conjecture. A ribbon is quasipositive if and only if, for some $n$, it is ambient isotopic to a subribbon of $\widetilde T_{n,n}$. This would characteriz…

Let $\Sigma$ be a link cobordism, and let two movies of $\Sigma$ have corresponding source and target link diagrams that are isomorphic. Each movie induces a map on the homology gr…

Let $F_k$ be the free group of rank $k$. For a group $G$ and a normal subgroup $N\lhd G$, let $V_k(G,N)$ be the set of $k$-tuples that generate $N$ as a normal subgroup of $G$, and…

Let $Y$ be a toroidal rational homology $3$-sphere. The refined toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 6$, then $Y$ is not $SU(2)$-abelian. This strengthen…

Let $Y$ be a toroidal rational homology $3$-sphere. The toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 4$, then $Y$ is not $SU(2)$-abelian. This conjecture predict…

Let $T(\mathbb{R}^d)$ be the tensor algebra, let $V$ be the space of loop invariants, and let $[V,\mathbb{R}^d]$ and $[T(\mathbb{R}^d),\mathbb{R}^d]$ denote the corresponding commu…

Let $N$ be a positive integer, let $T^2$ be the torus, and let $\mathrm{SkCat}_{\mathrm{SL}_N}(T^2)$ and $\mathrm{SkAlg}_{\mathrm{SL}_{N/d}}$ denote the corresponding skein categor…

Spherical-achirality conjecture. There are infinitely many spherically achiral knotoids that are planarily chiral.

Let $\zeta(u):S^1\sqcup S^1\to S^3$ be the standard link of linking number $u$, and let $\sigma\zeta(u)$ denote the link obtained from $\zeta(u)$ by the operation $\sigma$ using a…

Circle-pattern homeomorphism conjecture. The space of circle patterns is homeomorphic to the Teichmüller space of the closed torus.

Let $\mathfrak{m}(X_8,B^2)$ be the component of the immersion complex consisting of immersed annuli of figure-8 type in the disk $B^2$. Figure-8 disk fundamental-group conjecture.…