42 problems

Let ${\mathbb H}^1$ be the three-dimensional sub-Riemannian Heisenberg group. A Pansu sphere is the surface obtained as the union of all sub-Riemannian geodesics of a fixed curvatu…

Let $\alpha\in(0,1)$ and let $k>n$. An $\alpha$-Hölder continuous embedding is an embedding $F:\mathbb{B}^k\subset\mathbb{R}^k\to\mathbb{H}_n$ satisfying a Hölder estimate of expon…

Let $p$ be a prime and let $H_p$ denote the $p$-discrete Heisenberg group. Its Ekedahl invariants are the invariants $\operatorname{e}_i(H_p)$ associated with the classifying space…

Let $H^n$ be the Heisenberg group with $n>1$, and let … The critical exponent is … where $Q$ is the homogeneous dimension. A divergence symmetry is a Lie point symmetry whose actio…

Let $\mathbb H^1$ denote the first Heisenberg group. An intrinsic graph is a graph expressed using the intrinsic horizontal structure of $\mathbb H^1$, and stability refers to nonn…

Let $\mathbb H^1$ be the first Heisenberg group. An intrinsic graph is a graph with respect to a horizontal direction, and stability means nonnegative second variation of the $H$-p…

Multiplicity conjecture. For each $M\in\mathbb{N}_0$, there exists at least one distinct solution $u_M$ of problem $$ having at most $M$ modes.

Gath's conjecture. The optimal order of the error term is $2q-1$; that is, the infimum of the exponents $\kappa$ for which $|\mathcal E_q(t)|\lesssim t^\kappa$ holds equals $2q-1$.…

Refined Malle conjecture for ${\rm Heis}_4$. The group ${\rm Heis}_4$ fails to satisfy independence of local events, and

Let $p$ be in a suitable range with $p\gg 2$, and let denote the main square-function inequality for the operators considered in the paper. Square-function conjecture. The inequali…

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…

Metric-measure and Heisenberg-group conjecture. Problem $$ should yield the same result when considered over a metric measure space or a Heisenberg group.

Let $\mathbb{H}^1$ denote the first Heisenberg group. A $C^1_{\mathbb{H}}$-regular 2-dimensional submanifold with boundary is a 2-dimensional submanifold with boundary whose regula…

Curvature-exponent conjecture. Among all sub-Finsler Heisenberg groups, only the sub-Riemannian one has

Curvature exponent five conjecture. The metric measure space $(\mathbb{H},\mathsf d,\mathscr{L}^3)$ satisfies $\mathsf{MCP}(0,5)$ if and only if the reference norm is the $\ell^2$-…

Let $H_{2m+1}(\mathbb Z)$ denote the $(2m+1)$-dimensional discrete Heisenberg group. Conjecture. Higher $H_{2m+1}(\mathbb Z)$ have decidable Rational Subset Membership. If true, th…

Let $\mathbb H^1$ be the first Heisenberg group equipped with its Carnot–Carathéodory metric. A map $f:\mathbb R^2\to\mathbb H^1$ is a $\gamma$-Hölder embedding if it is an embeddi…

Let $q\geq 3$ be an integer, let $N_q(x)$ count lattice points in the Heisenberg dilation of the $(2q+1)$-dimensional unit Cygan–Korányi ball, and define … where…

Heisenberg spherical maximal conjecture. The same $L^p$ boundedness statement should hold when $n=1$; that is, ${\mathfrak M}$ should be bounded on $L^p({\mathbb H}^1)$ if and only…

Heisenberg lattice-point error exponent conjecture. For every $\epsilon>0$,

Let $H_{2l+1}$ denote the $(2l+1)$-dimensional Heisenberg group, and let $\mathcal{D}_{H_{2l+1},S}^{fin}(n)$ and $\mathcal{D}_{H_{2l+1},S}^{lin}(n)$ denote its weakly sofic and lin…

Let $f\colon\mathbb{R}^3\to\mathbb{R}$ be smooth and compactly supported, and let $p\in(1,2]$. In the Heisenberg group setting, write $D_{\mathsf{v}}^t$ for the vertical difference…

Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\operatorname{vol}(\mathcal{B})x^{2q+2}$, where…

Classification conjecture. The only smoothly bounded and non-characteristic domains of the Heisenberg group $\mathbb{H}^{1}$ are those diffeomorphic to the generalized solid torus…

Let $A\subset\mathbb{R}$ and let $\mathcal{H}$ be the Heisenberg group of $3\times3$ unipotent matrices. Define … The product set is…