5 problems

Let $X$ be a closed, oriented, smooth four-manifold with $b_2^+(X)>0$. Let ${\mathfrak{s}}={\mathfrak{s}}_0\otimes L_1$ be a $\operatorname{Spin}^c$ structure defined by…

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…

Let $X$ be an oriented smooth four-manifold, let ${\mathfrak t}$ be a spin$^u$ structure, and let ${\mathfrak s}$ be a spin$^c$ structure such that the Seiberg–Witten moduli space…

Let $\ell$, $F$, $L_1$, $\mathfrak{s}_0$, $SW(\mathfrak{s}_0\otimes L_1)$, and $Q_X$ denote the level, bundle and line-bundle data, Seiberg–Witten invariant, and intersection form…

Let $X$ be a four-manifold, let $E$ be a rank-two bundle, and let $z\in\mathbb A(X)$. Let $n_a$, $d_a$, $\overline{\operatorname{Red}}(W,E)$, $\overline V(z)$,…