73 problems

Let $Y$ be a closed, oriented three-manifold and $K$ a knot whose exterior is boundary-irreducible. A purely cosmetic surgery is a pair of distinct surgery slopes on $K$ producing…

Let $M$ be a manifold diffeomorphic to $S^3$, equipped with a Riemannian metric. An embedded minimal two-sphere in $M$ is an embedded minimal submanifold diffeomorphic to $S^2$. S.…

Let $(M^3,g)$ be a complete, non-compact Riemannian manifold. Say that $(M^3,g)$ is Ricci pinched when it satisfies the Ricci-pinching condition used in Hamilton's conjecture. Hami…

Let $(M^3,g)$ be a three-dimensional complete manifold with Ricci curvature $\operatorname{Ric}\geq 0$, and let $B_p(R)$ denote the geodesic ball of radius $R$ centered at $p\in M$…

Let $K$ be a knot with Alexander polynomial $\Delta_K\neq1$, let $F_K(x,\tau)$ be its knot-complement invariant, and let $a_{-m}(K;q)$ denote inverted Habiro coefficients. Inverted…

Dehn-surgery conjecture. The manifold at infinity of the even subgroup of $\Delta_{3,4,\infty;n}$ is the hyperbolic $3$-manifold obtained by Dehn surgery on the first cusp of…

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the…

Let $M_j$ be a sequence of Riemannian manifolds diffeomorphic to a $3$-torus such that … where … If the scalar curvature $R(M_j)$ satisfies $R(M_j)\geq -\frac{1}{j}$, then Sormani'…

Rubinstein's conjecture. $H$ is either isotopic to a minimal surface or isotopic to the oriented double cover of a non-orientable minimal surface with a vertical handle attached.

Let $K$ and $K'$ be knots, and let $S^3_0(K)$ and $S^3_0(K')$ denote their 0-surgeries. Akbulut–Kirby conjecture. If $K$ and $K'$ have homeomorphic 0-surgeries, then $K$ and $K'$ a…

Let $\mathbb{B}$ be the unit ball and let $\Delta\subset\mathbb{B}$ be the properly embedded one-manifold given in the proof of Theorem. Let $\overline{B}$ be a smooth compact Riem…

Let $(M,g(t))$ be a solution to the Ricci flow on a compact $3$-manifold with arbitrary initial metric. Hamilton's Harnack-estimate conjecture. A Harnack-type estimate holds for th…

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitt…

Hypertight-contact-structure conjecture. Every closed, orientable manifold that carries a taut foliation also carries a hypertight contact structure.

Orbit lattice conjecture for platycosms. All 10 platycosms have the orbit lattice property.

Spectral sequence differential conjecture. The spectral sequence collapses after the $E_3$ stage, and its differential

Let $M$ be an oriented connected closed three-manifold, possibly equipped with a banded trivalent colored graph, and let $p$ be a prime. Define … For a connected three-manifold $M$…

The generalised Casson conjecture. For every $\alpha\in\mathcal{A}(\mathcal{T},\mathbf{s})$,

The positive spun triangulation conjecture for Vol3. For any embedded closed geodesic $\gamma$, the pair $(N,\gamma)$ does not admit a positive spun triangulation.

Let $(M^3,g)$ be a complete non-compact Riemannian manifold, let $q\in M$, and suppose that $Rc\geq 0$ and $M$ has maximum volume growth. Writing $R$ for the scalar curvature and…

Let $M^3$ be a three-manifold that has an exhaustion by solid tori and admits a complete metric with scalar curvature $R>0$ and bounded geometry. The solid-torus exhaustion problem…

Let $M^3$ be the interior of a genus-$\gamma$ handlebody, equipped with a complete metric whose scalar curvature satisfies $R>0$. The handlebody positive-scalar-curvature conjectur…

Let $(M^3,g)$ be a contractible $3$-manifold equipped with a complete metric whose scalar curvature satisfies $R>0$. The contractible three-manifold positive-scalar-curvature conje…

Let $F$ be a smooth vector field on $S^3$ with fixed points $\{x_1,\ldots,x_n\}$, and let $H$ be a one-dimensional flow-invariant set such that each component $C$ of…

Let $p,q$ be positive integers with $\gcd(p,q)=1$, so that $Th(p,q)$ is a Turk’s head knot, and let $D_{2q}$ denote the dihedral group of order $2q$. Symmetry-group conjecture. … T…