28 problems

Solvmanifold deformation conjecture. (i) Every left-invariant complex structure on $G$ is Stein, respectively biholomorphic to ${\bf C}^n$ in the nilpotent case. (ii) Every small d…

Let $(M,\omega)$ be a closed symplectic manifold, and let $(M,\omega)\rightarrow P\rightarrow B$ be a Hamiltonian fiber bundle, meaning that its structural group can be reduced to…

Let $(M,J)$ be a complex manifold. A $p$-Kähler structure on $(M,J)$ is a closed, pointwise transverse $(p,p)$-form. Alessandrini–Bassanelli conjecture. If $(M,J)$ admits a $p$-Käh…

Let $W=N/\Lambda$ be a connected nilmanifold, let $Y=\pi(H)$ be a connected subnilmanifold of $W$, where $H$ is a connected closed subgroup of $N$ and $\pi:N\to W$ is the quotient…

Let $(G/\Gamma,\mu_{G/\Gamma},R_\alpha)$ be an ergodic nilsystem, and let $k_1,\ldots,k_\ell\in\mathbb Z$ be nonzero and distinct. For functions…

Brown–Graham–Landman conjecture. If $S$ is not a set of $d$-nilBohr recurrence for some $d$, then $S$ is not $2$-large.

Leibman's conjecture. The orbit closure

Let $\mathcal{M}$ be a compact, connected manifold, and let $X$ be a vector field on $\mathcal{M}$. The vector field $X$ is cohomology free if the associated cohomological equation…

Let $(X=G/\Gamma,\mu_X,T)$ be a connected nilsystem, with $T$ induced by left translation by $\tau\in G$. Write $G=G_0\Gamma$, where $G_0$ is the identity component, and write…

Conjectured optimal convergence criterion. Under this assumption, the averages

Let $G$ be a finite additive group, let $ta > 0$, let $s eq 1$, and let $f \colon G \to \mathbb{C}$ be a $1$-bounded function with $\|f\|_{U^{s+1}(G)} \geq \eta$. A degree $s$ filt…

Let $M$ be a 2-step nilmanifold with abelian complex structure. Suppose that there exists an ascending basis … such that $d\omega^{n+1}$ is non-degenerate. Let $\Gamma_1\in H^2(M)$…

Let $s\in\mathbb{N}$, let $G/\Gamma$ be an $s$-step nilmanifold, where $G$ is a connected, simply connected $s$-step nilpotent Lie group and $\Gamma$ is a cocompact discrete subgro…

Let $M$ be a compact nilmanifold and let $g$ be an Einstein metric on $M$. The Einstein metric conjecture. The metric $g$ is flat, and $M$ is a torus. This concerns the possible Ei…

Let $\rho$ be a homogeneous $\mathbb R^k$ action on a finite-volume space $G/D$, where $G$ is a connected Lie group and $D$ is a closed subgroup of $G$. An $r$-step nilmanifold is…

Nilsequence structure conjecture. There exist a nilmanifold $Y=G/\Gamma$, a continuous function $F\in C(Y)$, a point $y\in Y$, elements $a_1,\ldots,a_k\in G$, and a sequence…

Nilmanifold deformation conjecture. Then $(M^n,g)$ can be deformed to a complex nilmanifold $(N^n,h)$ of this form. Moreover, the step of $G$ is at most two, the left invariant com…

Let $t\in\mathbb{N}$, let $G$ be a nilpotent Lie group of step $t$, let $\Gamma$ be a cocompact lattice in $G$, and let $F\colon G/\Gamma\rightarrow\mathbb{C}$ be continuous. Let…

Let $s\geq 1$, and let $R\subset\mathbb N$ be a set of $s$-recurrence for $s$-step nilsystems. Recurrence transfer conjecture. Then $R$ is a set of $s$-recurrence for all $t$-step…

The expanding-map lifting conjecture. The infra-nilmanifold $\Gamma \backslash G$ admits an expanding map if and only if the nilmanifold $N \backslash G$ admits an expanding map.

Inverse conjecture for the Gowers norms. The stated correlation with a bounded Lipschitz nilsequence should hold for every $s$, $\delta$, $N$, and $f$ satisfying the hypotheses.

The conjecture from the 1960s. If there is an Anosov diffeomorphism $f$ on $M$, then some finite cover of $M$ is a nilmanifold. Furthermore, lifting $f$ to the finite cover yields…

Let $N$ be a nilpotent Lie group of step $r$ with Lie algebra $\mathfrak N$, let $\mathfrak N_j=[\mathfrak N,\mathfrak N_{j-1}]$ for $j=1,\dots,r$, let $\Gamma\setminus N$ be a com…

Console–Fino conjecture. These maps of complexes are quasi-isomorphisms, equivalently,

Let $M$ be a compact nilmanifold admitting a locally conformally Kähler (LCK) structure, and let $H(n)$ be the generalized Heisenberg group. Structure conjecture. Up to covering,…