6 problems

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$. A closed hypersurface encloses half the volume of $M$ when one of the regions it bounds has volume equal to…

Let $\Omega\subset\mathbb{R}^n$ be a connected domain with piecewise smooth boundary having $\theta$-corners, where $\theta\in(0,\pi)$. Fix $\eta>0$. For all $p\geq p_0(\Omega)$,…

Let $n>k\geq1$, let $F:X^p\rightarrow\mathcal{Z}_k([0,1]^n,\partial[0,1]^n;\mathbb{Z}_2)$ be a continuous family, and let $r\in(0,\tfrac12)$. There should exist a family…

Let $\Omega\subset\mathbb{R}^n$ be a connected domain whose piecewise smooth boundary has $\theta$-corners, with $\theta\in(0,\pi)$, and let $0\leq k<n$. Let $X^p$ be a $p$-dimensi…

Let $(M,g)$ be a closed Riemannian manifold, and let $\{\omega_k(M,g)\}_{k\in\mathbb N}$ denote its volume spectrum. Gromov's Weyl law conjecture. The volume spectrum satisfies a W…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, and let $\omega_p(M,g)$ denote its $p$-width. The constant $a(n)$ depends only on $n$. Gromov's Weyl law conjecture. The volume s…