21 problems

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…

Sharp $L^p$ boundedness conjecture. For general $\Lambda$, the operator $M^\Lambda$ is bounded on $L^p$ whenever

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…

Let $m\ge2$, and consider the theorem's lattice-point estimate for a step-two nilpotent group with a rational matrix in the sense that, for some $c>0$, its dilated matrix … is an i…

Let ${\mathbb H}^d\cong\mathbb{R}^{2d}\times\mathbb{R}$ be the Heisenberg group, and for $A=1$ and $\alpha=4$ let … The corresponding lattice-point discrepancy has a correct order…

Let $\mathfrak{n}$ be the generalized Heisenberg Lie algebra and let $\operatorname{ram}(\check{\mathrm{K}}^2)$ denote the ramification locus of the squared operator…

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.

Hypoellipticity conjecture. The Vladimirov sub-Laplacian $\mathscr{L}^{\alpha}_{sub}$ is a hypoelliptic operator on $\mathbb{G}$.

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 $G$ be a $k$-step nilpotent Lie group with $k\geq 2$, and let its isometry Lie algebra and Lie algebra of first integrals be denoted by the corresponding Lie algebras. First-in…

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 simply connected nilpotent Lie group with a simply transitive affine action arising from an étale affine representation $\rho:G\to\operatorname{Aff}(V)$. Let…

Rigidity conjecture. The theorem stated immediately after this assertion is true for all $k\geq 2$.

Quasi-isometric rigidity conjecture. Quasi-isometric simply-connected nilpotent Lie groups are isomorphic.

Let $G$ be a connected, simply connected nilpotent Lie group and let $(\pi,\mathcal{H})$ be an irreducible unitary representation of $G$ that is square-integrable modulo the center…

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…

Let $G$ be a nilpotent Lie group acting simply transitively on $\mathbb{R}^{n}$ by affine transformations. A central translation conjecture. Every such group $G$ contains a transla…

Corwin–Greenleaf conjecture. The algebra $\mathbb{D}(\mathfrak{g},\mathfrak{h},\lambda)$ is commutative if and only if

Let $N$) be a connected, simply-connected, two-step nilpotent Lie group, let $K$ be a compact group acting by automorphisms on $N$, and assume that $(K\ltimes N,K)$ is a Gelfand pa…