11 problems

Let $\mathbb{K}$ be a non-archimedean local field with ring of integers $\mathscr{O}_\mathbb{K}$, prime ideal $\mathfrak{p}=\textbf{p}\mathscr{O}_\mathbb{K}$, and residue field…

Let $(\mathcal{M}^{n},g(t))$ be a solution of Ricci flow such that $g(0)$ is sufficiently close, in a suitable norm, to a (locally) homogeneous metric on a quotient of a nilpotent…

Let $G$ be a simply connected, nilpotent Lie group of dimension $2n$ and let $J$ be a left-invariant complex structure on $G$. Hasegawa's biholomorphism conjecture. The complex man…

Lipsman's conjecture. The following two conditions are equivalent: (i) the $L$-action on $G/H$ is proper; (ii) the $L$-action on $G/H$ has the CI property.

Vladimirov sub-Laplacian conjecture. The Vladimirov sub-Laplacian of order $\alpha>0$ is a hypoelliptic operator on $G$ and is invertible on the space of mean-zero functions.

Let $\mathsf{G}$ be a connected nilpotent Lie group, and let $A$ be a continuous trace subquotient of $C^*(\mathsf{G})$, meaning a subquotient whose spectrum…

Let $G$ and $H$ be simply connected nilpotent Lie groups. Quasi-isometric rigidity conjecture. The groups $G$ and $H$ are quasi-isometric if and only if they are isomorphic. This w…

Largest Carnot subgroup conjecture. The set of $p$ for which $G$ is Markov $p$-convex is the same as the set of $p$ for which $H$ is Markov $p$-convex.

Let $N$ be a connected and simply connected nilpotent Lie group, and let $\Gamma\subseteq\operatorname{Aff}(N)$ act properly discontinuously and cocompactly on $N$; such an action…

Let $G$ be a 1-connected nilpotent Lie group, $H \subset G$ a closed subgroup, and $\Gamma \subset G$ a non-trivial discrete subgroup. Let $\mathcal{R}(\Gamma,G,H)$ denote the spac…

Let $G$ be a connected nilpotent real Lie group endowed with a left-invariant geodesic metric, and let $\widehat{\beta}_k$ denote its Diophantine exponent on $k$ letters. Assume th…