5 problems

Let $M$ be a closed hyperbolic $3$-manifold, and let $b_1(M)$ denote its first Betti number. Volume lower-bound conjecture. If … then … This conjectural lower bound is used to esta…

Let $B$ be the base of a surface bundle, let $p$ be a prime, and let $\kappa_i$ denote its Morita–Mumford classes. Vanishing conjecture. For $s\geq 1$ and $v\geq 0$, one has … The…

Let $M=M(e,g)$ be homeomorphic to an $\mathbb R^2$-bundle over a closed orientable surface of genus $g>1$, with Euler number $e=\langle[F],[F]\rangle$ given by the self-intersectio…

Let $S=S(g,p)$ be an orientable, connected surface of genus $g$ with $p$ punctures, and let $M$ be a hyperbolic surface bundle over $S^1$ with fiber of type $(-\chi,p)$, where…

Let $\pi\colon E\to B$ be an oriented surface bundle with closed fibers of genus $g$, equipped with a fiberwise metric. Let $\mathcal{H}(E)$ be the associated Hodge bundle, with fi…