6 problems

Let $M$ be a closed oriented $3$-manifold with volume form $\mu$, and let $X$ be an exact divergence-free vector field with $\omega=\iota_X\mu=d\alpha$. Its helicity is … A functio…

Let $X$ be a vector field in the setting of three-dimensional volume-preserving flows, and let its corresponding helicity level set be the set of vector fields with the same helici…

Rogers' universality conjecture. This central Lie algebra extension is universal.

Multi-intersection conjecture. Question has an affirmative answer provided there exist elements $\varphi_i$ lying in the identity component of the group of volume-preserving diffeo…

Helicity uniqueness conjecture. The helicity is the only Casimir function for $\operatorname{SDiff}(M)$.

Let $M$ be a manifold of dimension at least $3$, and let $\operatorname{Diff}_\mu(M)$ denote its group of volume-preserving diffeomorphisms equipped with the right-invariant $H^s$…