57 problems

Let $\mathcal{H}$ be the space of heteroclinic orbits of the semilinear parabolic equation, and let $Y_\lambda$ denote the ambient space in which these orbits are considered. Ident…

Fibred Dehn twist exact sequence conjecture. Under conditions which render $a$ and $b$ well-defined, they fit into a long exact sequence

Let $\Sigma$ be a surface and let $L,L'\subset\Sigma$ be disjoint circles. Write $\widehat{V}_L^n$ and $\widehat{V}_{L'}^n$ for the associated Lagrangian correspondences between th…

Owens' H-thin conjecture. For any H-thin knot $K$,

Floer-theoretic string topology operations conjecture. For every marked chord diagram $\Gamma$, there is an umkehr map

Equivariant Seiberg–Witten Floer and Heegaard Floer correspondence. For every such $Y$ and $\mathfrak{s}$, there are isomorphisms

General Floer homology and asymptotic growth conjecture.

Let $Y$ be a three-manifold with $b_1(Y)=1$ and a nontorsion $spin^c$ structure ${\mathfrak{s}}$. Let ${\text{SWF}}$ and ${\text{SWF}}_0$ be the associated periodic Seiberg–Witten…

Kronheimer–Mrowka's conjecture. These equivariant Seiberg–Witten Floer homologies are well-defined, meaning that they are independent of the particular choices of metrics and pertu…

Let $X_1$ be a cobordism between three-manifolds $Y_1$ and $Y_2$, and let $X_2$ be a cobordism between $Y_2$ and $Y_3$. Denote by $\mathcal{D}_{X_1}$, $\mathcal{D}_{X_2}$, and…

Skein–DT sheaf conjecture. There is an isomorphism

GJS conjecture.

Let $Y$ be a rational homology $3$-sphere. Call $Y$ $SU(2)$-abelian if every representation $\pi_1(Y)\to SU(2)$ has abelian image. Zhang's conjecture. If $Y$ is $SU(2)$-abelian, th…

Spectral-sequence non-collapse conjecture. There exists a strongly invertible knot $\widetilde{K}$ such that the spectral sequence of Theorem 1 collapses on the $E_k$ page for some…

Equivariant concordance conjecture. The number $s_\tau$ is an equivariant concordance invariant.

Let $M$ be an irreducible rational homology sphere, and let $HM_*(M)$ have a direct $\mathbb{F}$-summand, where $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. The Floer-summand conjecture. T…

The algebraically tight contact-pair conjecture. The pair $(\xi_-,\xi_+)$ is homotopic through contact pairs to a positive contact pair.

Let $(M,\omega)$ be a symplectic manifold of dimension $2d$ with $2c_1(M,\omega)=0$, and let $L_1,\dots,L_n$ be graded Lagrangian spheres. Let $\mathcal{A}$ be the directed Fukaya…

Let a 4d $\mathcal{N}=2$ theory be reachable from 4d $\mathcal{N}=4$ super-Yang–Mills theory by an RG flow, and let $M_3$ be a 3-manifold. Spectral-sequence conjecture. The RG flow…

Let $M_3$ be a closed 3-manifold, and let $\mathcal{H}_{\mathrm{VW}}(M_3)$ denote its Vafa–Witten homology. Infinite-dimensionality conjecture. The space…

Suppose that $Y$ is a closed oriented $3$-manifold, and $E$ is a $U(2)$ bundle with admissibility condition $w=c_1(E)\in H^2(Y;\mathbb{Z})$ a non-torsion odd class. Choose a spin-…

Let $Y$ be a closed oriented $3$-manifold. Kronheimer–Mrowka's conjecture. The framed instanton homology and the tilde version of monopole Floer homology satisfy … This is a specia…

Sutured instanton–Heegaard Floer equivalence conjecture. For every balanced sutured manifold $(M,\gamma)$, there is an isomorphism

Arnold's conjecture. The number of fixed points of $f$ satisfies

Juhász's conjecture. The sutured manifold $(M,\gamma)$ admits a taut foliation of depth at most $2k$.