16 problems

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

Quadratic growth conjecture.

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

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 $H_{2l+1}$ 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 $\mathcal{A}=\{\mathsf{R},\mathsf{C},\mathsf{H},\mathsf{O}\}$ be the collection of normed division algebras. For ${\mathsf K}\in\mathcal{A}$ and $n\in\mathsf N$, let…

Let $\mathbb H_n$ be the Heisenberg group, let $\gamma^{s}_{\mathbf{X},L^2}(\mathbb H_n)$ and $\gamma^{(s)}_{\mathbf{X},L^2}(\mathbb H_n)$ be the corresponding sub-Laplacian Gevrey…

Let $\mathbb{Q}[\varepsilon_n]=\mathbb{Q}[x]/(x^n)$, and let $\mathbf{H}[\varepsilon_n]$ denote the group scheme obtained from the Heisenberg Lie algebra tensored with…

Let $R$ be a ring which is torsion-free and finitely generated over $\mathbb{Z}$, and let $\mathbf{H}(R)$ be the corresponding Heisenberg group. Write $\alpha(\mathbf{H})$ for the…

Sharpness and optimizer conjecture. There exists a sharp constant $C>0$ such that

Let $K$ be a number field, let $g$ be the number of primes of $K$ above a rational prime, and let $(\mathbf{e},\mathbf{f})\in\mathbb{N}^g\times\mathbb{N}^g$ satisfy … Here…

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…

Asymptotic sharpness conjecture. There exists a prime $q$ such that, for every prime $p \geq q$, there exists a $\frac{p+1}{2}$-dimensional module for $H_1$ over a field of charact…

Let $\mathcal{H}^3$ and $\mathcal{H}^5$ denote the three- and five-dimensional Heisenberg groups. The area distortion of $\mathcal{H}^3$ in $\mathcal{H}^5$ measures the filling are…