12 problems

Let $M_g^2$ be a surface in ${\mathbb T}^3$, let $B$ be a direction, and let ${\cal F}_B(M_g^2)$ denote the corresponding foliation or section system. Call ${\cal F}_B(M_g^2)$ chao…

Let $f$ be a quasiperiodic function, and let $b_f^k(c)$ denote the density of its $k$-th Betti number at level $c$, defined by averaging the corresponding Betti number over expandi…

Let $M$ be a compact manifold and let several closed $1$-forms on $M$ be given. Their common level is the intersection of the corresponding level sets at a generic choice of levels…

Let $M_Fcsubset\mathbb T^3$ be a connected smooth two-manifold homologous to zero, and let a magnetic field determine trajectories on it. For a smooth one-parameter family…

In a periodic energy surface, let the level-set trajectories induced by a magnetic field include strongly chaotic trajectories, meaning trajectories whose lifts to the plane have n…

Let $m=4$ and $s=1$, and fix a generic Fermi level $\epsilon=const$ in the space $T^4$. Consider the stably integrable set in the Grassmannian $G_{2,2}$ of pairs $(l_1,l_2)$. Full-…

Let $m=1$ and $s=1$ denote the physical case, and consider the set of chaotic cases for the corresponding quasiperiodic dynamics. Hausdorff-dimension conjecture. The Hausdorff dime…

Let $\varepsilon_c$ be a level hypersurface of a smooth quasiperiodic function on ${\mathbb T}^4$, and let $\Pi\in G_{4,2}(\mathbb R)$ be a $2$-plane. The level is topologically co…

Let $f:{\mathbb T}^3\to\mathbb R$ be a function, and let ${\cal E}(f)$ be its exceptional set of directions, namely the directions not belonging to the stability-zone set. Novikov'…

Let $M_g^2\subset{\mathbb T}^3$ be a generic embedded surface, let $\ell$ be a soul label, and let ${\cal D}_{\ell}(M_g^2)$ denote its stability region; write $\|\ell\|$ for the no…

Let $M$ be a compact manifold and let $\omega$ be a closed $1$-form on $M$ with singular points. A leaf means a connected component of a level set of a local primitive of $\omega$…

Let $Fto oalign{vskipbaselineskip}$ be a generic smooth function on the three-torus, and let ${cal E}F$ denote the exceptional set of directions associated with its quasiperiodic…