16 problems

Naive bilinear Kakeya conjecture. Let $\varepsilon>0$. For every $\delta\in(0,1)$ and every pair of transversal families of Heisenberg $\delta$-tubes…

Let $(\mathbb H,d^{\mathbb H})$ be the Heisenberg group equipped with its Carnot–Carathéodory metric. Lee–Naor conjecture. The metric space $(\mathbb H,d^{\mathbb H})$ does not adm…

Distance-realization property. Then the distance $\operatorname{dist}(P,S_R)$ is realized by the point

Let the bilinear tube estimate in Theorem $1$ be understood with its stated $L^p$ exponent, and let the sharp exponent mean the smallest exponent for which that estimate holds. Sha…

Let $H$) be the Heisenberg group, and let $H\rtimes_{\psi}\mathbb{R}$ be its semidirect product with $\mathbb{R}$, where the action is given by … An embedding is $\varepsilon$-thic…

Let $J$ denote the dilational momentum and consider a zero-energy orbit meeting the $z$-axis. Bifurcation conjecture. If … then the orbit has self-similarity like a generic $H=0$ o…

Let $H$ be the Kepler–Heisenberg Hamiltonian, and let $z(t)$ denote the vertical coordinate of a zero-energy orbit. An orbit avoids the $z$-axis if it never meets the set where the…

Let $\Omega$ be an open subset of $\mathbb{R}^k$, let $n$ be a positive integer, and let $C^{\gamma}(\Omega,\mathbb{H}_n)$ denote the maps that are $\gamma$-Hölder with respect to…

Uniqueness conjecture. The shifted Heisenberg spheres above are the only closed minimizers for $E_1$ with zero energy in $\mathcal{H}_1$. The source establishes that these spheres…

Let $H(\mathbb R)$ be the Heisenberg group with a Carnot–Carathéodory metric arising from a norm on the horizontal plane. In the sub-Riemannian case, Pansu's bubble set is the uniq…

Let $(H(\mathbb R),d_Q,\lambda)$ be the Heisenberg group equipped with Haar measure $\lambda$ and an arbitrary Carnot–Carathéodory metric $d_Q$ arising from a norm on the horizonta…

Let $H(\mathbb R)$ be the Heisenberg group with its sub-Riemannian Carnot–Carathéodory metric. Let $S$ be a surface enclosing a region $E$, let $\lambda$ denote Haar measure, and l…

Let $\mathbb{W} \subset \mathbb{H}$ be a vertical subgroup with complementary subgroup $\mathbb{V}$, and let $\phi \colon \mathbb{W} \to \mathbb{V}$ be an intrinsic Lipschitz funct…

Consider the Heisenberg-group dynamical system described above, with initial conditions and orbits under its Hamiltonian flow. An orbit is asymptotic to a helix if it approaches a…

Let $\mathbb H$ be the discrete Heisenberg group, let $B_n$ be its radius-$n$ ball with the word metric $d_W$, and let $c_1(B_n,d_W)$ denote the least distortion of an embedding of…

Coarse total vertical perimeter conjecture. For every measurable $A\subseteq\mathbb H(\mathbb R)$ and every $\varepsilon\in(0,1/2)$,