22 problems

On a hyperhermitian manifold $(M,I,J,K,g)$, let $4\Omega=\omega_J-i\omega_K4$, where $4\omega_J4$ and $4\omega_K4$ are the fundamental forms associated with $J$ and $K$, and let…

Let $(M^{4n},I,J,K,\Omega)$ be a compact HKT-manifold, where $I,J,K$ are the quaternionic complex structures and $\Omega$ is its HKT-form. Let $F\in C^\infty(M,\mathbb{R})$ be a sm…

Alekseevsky–Marchiafava–Pontecorvo conjecture. The only complete simply connected hHqK manifold with negative scalar curvature is $\mathbb{H}H^n$.

Alekseevskii's classification conjecture. The homogeneous quaternionic spaces consist of compact symmetric quaternionic spaces and normal quaternionic spaces.

The six equiangular lines are represented by vectors in quaternionic two-space. Regular-720-cell conjecture. The $720$ vectors in $\mathbb{H}^2$ giving the six equiangular lines ar…

Quaternionic Haiman conjecture. There is a $(g+1)^n$-dimensional quotient ring of the zero-fiber ring of $W$.

Let $m\geq 2$. Consider the cohomogeneity-one action with group triple … Its principal orbit is $\mathbb{S}^{4m+3}$, identified with the total space of the quaternionic Hopf fibrat…

Let $(\mathbb{H}P^{N},g_{0})$ be quaternionic projective space with its canonical metric. The identity-map p-energy conjecture. The identity mapping … minimizes $p$-energy in its h…

Let $\mathfrak{H}$ be the quaternionic Hilbert space under consideration, and let $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_d)$ and $\boldsymbol{r}=(r_1,\dots,r_N)$ be positiv…

Berndt–Tamaru conjecture. These were all the possible subspaces with constant quaternionic Kähler angle.

Consider the quaternionic analogue of the projective-design problem associated with Zauner's conjecture: the existence of a tight projective $2$-design, also known in the complex s…

A maximal quaternionic simplex is a simplex in $\mathbb{H}\mathbb{P}^{d-1}$ attaining the maximal size permitted by the relevant simplex bound. The only known example is the simple…

Let $f:S^2\to\textbf{HP}^n$ be a proper minimal immersion, meaning that $f(S^2)$ is not contained in any totally geodesic submanifold of $\textbf{HP}^n$, and suppose that $f$ has c…

Let $(S,[D]_c,\mathcal{L},\nabla)$ be the data used in the generalized Feix–Kaledin construction, and let $M$ be the resulting manifold. Let $k$ be the dimension of the subalgebra…

Equivariant quaternionic splitting conjecture. There is a $V_4$-equivariant stable splitting of $\operatorname{Sp}(V_{\mathbb H};W_{\mathbb H})$ into Thom spaces of finite-dimensio…

Let $t\in\mathbb{N}^*$, let $S^3\subset\mathbb{R}^4$ be the unit $3$-sphere, and let $\nu(5^t)$ denote the integer solutions of … For $x\in\nu(5^t)\cup\nu(5^{t-1})$, write…

Identify $\mathbb{R}^4$ with the quaternion skew field $\mathbb{H}$. Let an analytic surface in $\mathbb{R}^4$ have two noncospheric circular arcs through every point, fully contai…

Nonexistence conjecture. There does not exist a tight simplex of $14$ points in ${\mathbb H}{\mathbb P}^2$.

Local dimension conjecture. There exists a $12$-point, respectively $13$-point, tight simplex in ${\mathbb H}{\mathbb P}^2$ such that, in a neighborhood of it, the space of tight s…

Existence conjecture. As $d\to\infty$, there exist tight $N$-point simplices in ${\mathbb H}{\mathbb P}^{d-1}$ for every $N$ satisfying

Quaternionic Calabi–Yau conjecture. On a compact HKT manifold with $SL(n,\mathbb H)$-holonomy of the Obata connection, there exists a balanced HKT metric; if it exists, it is uniqu…

Let $M$ be a real hypersurface in $SU_{2,m}/S(U_2U_m)$ whose principal curvatures satisfy the exceptional conditions in case (iv) of the main classification theorem: its normal bun…