98 problems

Isomorphism conjecture. The map $\varphi$ is an isomorphism of Poisson vertex algebras.

A closed hyperbolic four-manifold is a closed four-dimensional manifold equipped with a hyperbolic structure. Its Seiberg--Witten invariants are the invariants associated with its…

Let $Y_1$ be a cobordism from $\Sigma_1$ to $\Sigma_2$ and $Y_2$ a cobordism from $\Sigma_2$ to $\Sigma_3$, with adapted harmonic Morse functions, compatible…

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$ and odd $b^+(X)-b_1(X)$. A Seiberg–Witten basic class is a cohomology class associated wi…

Let $N$ be a homology sphere and let $g$ be a generic metric on $N$. Write $\chi_{SW}(N,g)$ for the Seiberg–Witten invariant, $\mathbf{F}(N,g)$ for the corresponding invariant appe…

In Example 1, let $Y$ be a closed connected $3$-manifold with a Morse function having a unique maximum and minimum, and let $\underline{Y}$ and $\underline{f}$ be the associated cy…

Let $X$ be a smooth spin rational homology $S^1\times S^3$ which is oriented and homology oriented, and let $-X$ denote $X$ with reversed orientation but the same homology orientat…

Involutive Heegaard Floer–Seiberg–Witten correspondence. For every rational homology sphere $Y$ with a self-conjugate $\operatorname{spin}^c$ structure $\mathfrak{s}$, there is an…

Let $X$ be an oriented 3-manifold with $b_1(X)>0$, and let $I$ denote the invariant considered in the paper. Let the Seiberg–Witten invariant of $X$ be given in the relevant parame…

Canonical basic class conjecture. Any manifold obtained this way should have the canonical class, up to sign, as its only Seiberg–Witten basic class.

Lagrangian matching invariants conjecture. The invariants coincide with the Seiberg–Witten invariants of $X$ and are, in particular, independent of the chosen broken fibration.

Let $Y$ be a closed oriented $3$-manifold equipped with a fibration $f:Y\to S^1$ of fiber genus at least $2$, let $h\in H_1(Y;\mathbb{Z})$, let $c\in H^2(Y;\mathbb{R})$ evaluate po…

CMT–CF comparison conjecture. For all sufficiently large $L$, there is an isomorphism of $R[U]$-modules

Heegaard–Monopole–Taubes comparison conjecture. There are isomorphisms of $R[U]$-modules

Seiberg–Witten Taubes orbit counting conjecture.

Heegaard Floer–Seiberg–Witten Floer equivalence conjecture. There are isomorphisms of $R[U]$-modules

Heegaard Floer homology and Seiberg--Witten monopole Floer homology are two constructions of Floer-theoretic invariants associated with three-manifolds. Isomorphism conjecture. The…

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

Let $X$ be a complex surface, let $C$ be a holomorphically embedded submanifold, and let $Z(X,C,g)$ denote the manifolds constructed from this data. For finitely many values of $g$…

The constructions described produce families of homeomorphic simply connected $4$-manifolds with identical Seiberg–Witten invariants. Nondiffeomorphism conjecture. No two of these…

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…

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

Let $X$ be a closed, oriented four-manifold with $b_1(X)=0$. Let $\eta\geq 0$ be an integer and let $z$ be a homology insertion satisfying … For a $\operatorname{spin}^u$ structure…

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 $X$ be a closed, oriented, smooth four-manifold with $b_1(X)\geq 0$ and $b_2^+(X)\geq 1$. Let $\mathcal{M}_{\mathfrak{t}}$ be a moduli space of $\operatorname{PU}(2)$ monopoles…