27 problems

Let $\varphi$ be a closed $G_2$-structure, meaning a positive closed three-form, on a closed $7$-manifold. Consider the Laplacian flow … A closed $G_2$-structure has sufficiently s…

Let $M$ be a compact, locally irreducible Riemannian manifold that is rigidly scalar-flat, meaning that it admits a scalar-flat metric but no metric of positive scalar curvature. I…

A compact complex manifold $M$ of complex dimension $n$ has vanishing first Chern class, $c_1(M)=0$. A Hermitian connection is a connection preserving the Hermitian structure; it h…

Special-holonomy formality conjecture. Any compact manifold admitting a Riemannian metric with restricted holonomy strictly contained in $\mathrm{SO}(n)$ should be formal.

Let $M_\zeta$ be the ALE resolution constructed from the quotient $V/\Gamma$, and let $\omega_\zeta$ denote its Kähler form. The Calabi–Yau ALE metric conjecture. The ALE space…

A compact holonomy $\mathrm{G}_2$ manifold is a compact $7$-dimensional manifold admitting a metric whose holonomy is exactly $\mathrm{G}_2$. The known topological constraints incl…

In a Calabi–Yau $3$-fold or a $G_2$-manifold, generalized monopoles are critical points of the Yang–Mills–Higgs energy … The relevant invariants are expected to count generalized m…

Fix $\lambda<0$ and consider the 4-dimensional space of local $\textup{SU}(3)$-invariant shrinkers, together with the complete AC shrinkers among them on $\Lambda^2_-\mathbb{CP}^2$…

Let $\Lambda^2_-\mathbb{S}^4$ be the anti-self-dual bundle of the 4-sphere, and let the explicit AC shrinker be the complete $\textup{Sp}(2)$-invariant shrinker described in Exampl…

Let $\Lambda^2_-\mathbb{S}^4$ be the anti-self-dual bundle of the 4-sphere, and consider complete AC $\textup{Sp}(2)$-invariant Laplacian shrinkers on it. The explicit shrinker is…

The space of $\textup{SU}(3)$-invariant closed $\textup{G}_2$-cones is 2-dimensional. Consider the subset consisting of cones that occur as asymptotic cones of $\textup{SU}(3)$-inv…

Let $\Lambda^2_-\mathbb{CP}^2$ denote the anti-self-dual bundle of the complex projective plane. A complete expander has quadratic-exponential volume growth when its volume grows a…

Consider complete AC shrinkers with $\textup{SU}(3)$ symmetry on the relevant cohomogeneity-one $\textup{SU}(3)$-invariant geometry. SU(3)-invariant shrinker-finiteness conjecture.…

Let $\Lambda^2_-\mathbb{CP}^2$ be the anti-self-dual bundle of the complex projective plane. An AC expander or shrinker is a complete Laplacian-flow expanding or shrinking soliton…

Let $\Lambda^2_-\mathbb{CP}^2$ denote the anti-self-dual bundle of the complex projective plane, equipped with the natural $\textup{SU}(3)$-action. An expander is a Laplacian-flow…

The Hitchin index measures analytical contributions to the virtual dimension of moduli spaces associated with special geometric structures. The known analogy with stability indices…

Conjecture on negative quaternionic–Kähler manifolds. There are no closed simply-connected negative quaternionic–Kähler manifolds other than the known symmetric examples.

A quaternionic Kähler manifold is a Riemannian manifold whose holonomy is contained in $\operatorname{Sp}(n)\operatorname{Sp}(1)$; its scalar curvature is positive when its scalar-…

The families consist of the Spin(7) holonomy metrics $(M_{S^5},\Psi_{\mu})$ and $(M_{\mathbb{C}P^{2}},\Upsilon_{\tau})$, whose AC members are asymptotic to the same Spin(7) cone. T…

A manifold of special holonomy is a Riemannian manifold whose holonomy group is a special holonomy group. A space is formal when its rational homotopy type is determined by its coh…

A compact Ricci-flat manifold is a compact manifold with a Ricci-flat Riemannian metric, and it is stable when it has no relevant instability; a Riemannian manifold has special hol…

Consider the supersymmetric warped compactifications and corresponding generalised structures described in the paper, including the AdS backgrounds whose cones are viewed in…

Let $M$ be the internal space of a warped compactification of the form $\mathbb{R}^{D-1,1}\times M$, with supersymmetry encoded by a generalised $G$-structure in…

Let $Y$) be a Calabi–Yau $3$-fold with an antiholomorphic isometric involution, and let $Z=(Y\times S^1)/\hat{\mathord{{\Greekmath 0113}}}$ be the resulting quotient, whose singula…

Barely $G_{2}$ resolution conjecture. Each singular point of $Z$ should be resolvable using an ALE $4$-manifold with holonomy $SU(2)$, yielding a smooth manifold with holonomy…