11 problems

Auslander conjecture, special case. The fundamental group $\pi_1(M)$ contains a solvable subgroup of finite index.

Multiplicity-free holonomy conjecture. For every finite group $H$, there exists a Bieberbach group $\Gamma$ with $\mathbb{Q}$-multiplicity-free holonomy representation…

Orbit lattice conjecture for platycosms. All 10 platycosms have the orbit lattice property.

Let $(M,g)$ be a flat manifold and let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$. Generic-potential bounded-eigenfunction problem. For generic $V$, are the eigenfunctions u…

Let $g$ be a Riemannian metric on the torus $\mathbb{R}^n/\mathbb{Z}^n$, and suppose that its geodesic flow is toric integrable. Toric integrability conjecture. Then $g$ is flat. T…

Pontryagin-class non-vanishing conjecture. For the orientable flat manifolds $\widehat{\mathsf{LS}}_n$, the mod $2$ reduction of $p_i(\widehat{\mathsf{LS}}_n)$ is non-zero whenever…

Let $G$ be a non-trivial finite group and let … be a representation. Write $\rho^{\mathbb{R}}$ for the representation obtained by extending scalars to $\mathbb{R}$, and let…

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

Let $H>0$ and $g\in\mathbb{N}\cup\{0\}$. Consider the moduli space of non-congruent, connected, closed $H$-surfaces of genus at most $g$ in a fixed flat $3$-torus. Finiteness conje…

Let $n\geq 3$, and let $(M^n,\bar{g})$ be a connected closed flat Riemannian manifold. A metric has pointwise positive Q-curvature if its Q-curvature is positive at every point. Ri…

Let $(M,g)$ be a Riemannian torus, meaning a manifold diffeomorphic to a torus equipped with a Riemannian metric. A finite set $B\subset M\setminus\{x,y\}$ is a blocking set for po…