Suppose $(X, g)$ is a compact, spin Riemannian $7$-manifold, with Dirac operator${\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g : C^\infty (X, {\mathrm{S}\mspace{-10mu}/\mspace{2mu}}) \to C^\infty (X, {\mathrm{S}\mspace{-10mu}/\mspace{2mu}})$. Let $G$ be $\mathrm{SU}(m)$ or $\mathrm{U}(m)$, and $E \to X$ be a rank $m$ complex bundle with $G$-structure. Write $\mathcal{B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal $\mathbb{Z}_2$-bundle$O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E \to \mathcal{B}_E$ parametrizing orientations of $\operatorname{det} {\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} \, A}$ for twisted elliptic operators ${\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} \, A}$ at each $[A]$ in $\mathcal{B}_E$. A theorem of Walpuski [33] shows $O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E$ is trivializable.

We prove that if we choose an orientation for $\operatorname{det} {\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g$ , and a flag structure on $X$ in the sense of [17], then we can define canonical trivializations of $O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E$ for all such bundles $E \to X$, satisfying natural compatibilities.

Now let $(X, \varphi, g)$ be a compact $G_2$-manifold, with $d (\ast \varphi) = 0$. Then we can consider moduli spaces $\mathcal{M}^{G_2}_E$ of $G_2$-instantons on $E \to X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with $\mathcal{M}^{G_2}_E \subset \mathcal{B}_E$.The restriction of $O^{{\mathrm{D}\mspace{-12mu}/\mspace{4mu}}^g}_E$ to $\mathcal{M}^{G_2}_E$ is the $\mathbb{Z}_2$-bundle of orientations on $\mathcal{M}^{G_2}_E$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces $\mathcal{M}^{G_2}_E$.

This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of $G_2$-manifolds $(X, \varphi, g)$ by counting moduli spaces $\mathcal{M}^{G_2}_E$, with signs depending on a choice of orientation.

We use the construction of unfolded Seiberg–Witten Floer spectra of general $3$-manifolds defined in our previous paper to extend the notion of relative Bauer–Furuta invariants to general $4$-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.

We prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modeled on a half-space. To this end, we develop the theory of weak free boundary inverse mean curvature flow further and establish the monotonicity of a modified Hawking mass along this flow. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $\mathbb{R}^3$ conjectured by G. Huisken.

In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our argument also implies the denseness of the minimal hypersurfaces realizing min‑max widths for generic metrics. This partially supports the equidistribution of the minimal hypersurfaces realizing min-max widths conjectured by F.C. Marques, A. Neves, and A. Song in [19].