4 problems

Let $\varepsilon:\mathbb{T}^4\to\mathbb{R}$ be a smooth function, let $\varepsilon_c=\varepsilon^{-1}(c)$ be a level hypersurface, and let $\Pi\in G_{4,2}(\mathbb{R})$ be a $2$-pla…

Let $f:\mathbb{T}^3\to\mathbb{R}$ be a smooth function and let $\mathcal{E}(f)$ be its exceptional set of directions. Let $\dim_H$ denote Hausdorff dimension. Novikov and Dynnikov'…

Let $M_g^2\subset\mathbb{T}^3$ be a generic embedded surface, let $\ell$ be a stability-zone label, and let $\mathcal{D}_{\ell}(M_g^2)$ denote its stability region. Write…

Let $M$ be a manifold and let a collection of closed $1$-forms on $M$ be given. Their common level is a connected component of the simultaneous level set determined by local primit…