13 problems

Normal holonomy conjecture. The normal holonomy group is isomorphic to a subgroup of $\operatorname{SO}(n)$.

Let $M$ be a compact complex manifold of real dimension $2n$ with vanishing first Chern class. A hermitian structure on $M$ is a Riemannian metric compatible with its complex struc…

Full holonomy conjecture. For $d\geq 3$, $n>d$, and initial conditions with $D(P(0))$ not scalar (non-regular graph), the restricted holonomy group of Laplacian dynamics is

Let $(M,g)$ be a compact connected Riemannian manifold with reducible holonomy, and let $D$ be a conformal product structure on $(M,g)$ different from the Levi-Civita connection of…

Let $G/H$ be a reductive Lorentzian homogeneous space of dimension $m\ge 4$, with isotropy group $H$ acting indecomposably but non-irreducibly. A plane wave is a Lorentzian manifol…

Markus conjecture. Any connected closed flat affine manifold with parallel volume is complete.

Wilson-loop injectivity conjecture. If $(M,g)$ is a closed odd-dimensional Riemannian manifold with Anosov geodesic flow, then the Wilson loop operator

Let $M^n$ carry the metric … and density potential $\phi=x_1+x_2+\cdots+x_n$. Compute the holonomy Lie algebra at the origin $(0,0,\ldots,0)$. Strictly upper triangular holonomy co…

Guichard-Wienhard conjecture. There exists a generalized flag variety $X$ and a compact fiber bundle $M\to S$ such that $\overline{\rho}$ is the holonomy of a locally homogeneous…

Let $\cal F$ be a Riemannian foliation with bounded holonomy, and suppose that no leaf has infinite fundamental group. Let $H_p$ denote the holonomy group at $p$. Bounded-holonomy…

Let $\cal F$ be a Riemannian foliation, let $H_p$ denote the holonomy group at $p$, and let $\mathcal V_p$ be the corresponding vertical space. The inclusion…

Let $\mathcal{F}$ be a singular holomorphic foliation in $\mathbb{C}\mathbb{P}^2$. Let $L_1$ and $L_2$ be two non-invariant projective lines, and let…

Let $\Gamma$ be a lattice with trace field $K$, where $K$ is the field generated by the traces of elements of $\Gamma$. Let $h_K$ and $h_K^+$ denote the class number and narrow cla…