73 problems

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 $M$ 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 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 $(M^3,g)$ 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 $F$ 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…

The three-dimensional case of Lutz and Nevo's conjecture. For every flag triangulation $T$ of $S^3$ with $\gamma_2(T)=0$, there is a sequence of such collapses that reduces $T$ to…

Uniqueness conjecture. $T_{12}$ is, up to isomorphism, the only flag triangulation of $S^3$ with 12 vertices in which every edge is contained in a square.

Let $M$ be a complete finite-volume once-cusped hyperbolic three-manifold. An ideal triangulation of $M$ is a triangulation whose vertices lie at the cusp, and an ideal triangulati…

Lin geography restriction conjecture. The Heegaard Floer module $\operatorname{HF}^-(Y)$ satisfies the Lin geography restriction for all rational homology spheres $Y$.

The $L$-space conjecture. The following statements are equivalent:

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…

Let $M_3$ be a weakly negative plumbed three-manifold with defects labelled by highest weights $\nu_v\vec\omega$ at nodes $v\in V_W$, and let $\hat Z_b(M_3;W_{\{\nu_v\}};\tau)$ be…