12 problems

Let ${\mathcal{M}}_\B^{{\bm \mu},{\bf r}}$ be the generic wild character variety, let $d_{{\bm \mu},{\bf r}}$ be its dimension, and let…

Converse characterization conjecture. If $(\mathfrak{I}_{\hat\Psi},\mathfrak{t}_{\hat\Psi})$ is a commutative Frobenius algebra in

Finitely dominated aspherical Poincaré space manifold conjecture. Every finitely dominated aspherical Poincaré space is homotopy equivalent to a closed TOP manifold.

Mod-8 signature conjecture. The evaluation

Hyperbolicity conjecture. If $M$ is rationally hyperbolic, then $\overline{M}$ is $\mathbb{Z}/p^r$-hyperbolic for almost all primes $p$ and all integers $r\geq 1$.

Let $X^n$ be a space and let $p$ be a prime. Say that $H^*(X;\mathbb{Z}_p)$ is $k$-periodic up to degree $c$ when multiplication by a periodicity element induces successive isomorp…

Anomalous-module freeness conjecture. This phenomenon should generalize to $\widetilde{\operatorname{Gr}}_5(2^t)$ and $\widetilde{\operatorname{Gr}}_4(n)$ with

Let $(X,Y)$ be a Poincaré pair with orientation system $\mathcal{O}$ and fundamental class $[X]$, and let $\partial_*[X]$ denote the boundary class in $Y$. The two requirements are…

Extended Poincaré duality conjecture. Poincaré duality holds for $D_{\mathbf v(\mathbf w\varphi)^n}$: there is a perfect pairing of $D_{\mathbf v(\mathbf w\varphi)^n}$-modules

Shrinking-tubular-neighborhood conjecture. For a general M5-brane worldvolume in a general eleven-dimensional manifold, the mixing between $H^4(Y^{11}_r;\mathbb R)$ and…

Let $M^m$ be a closed, smooth, oriented Riemannian manifold and let $N^n$ be a closed submanifold of codimension $m-n \geq 2$. Define the compact Riemannian manifold … where…

Hain and Looijenga's conjecture. These intersection pairings are perfect (Poincaré duality), and $R^{\bullet}_c(M_g)$ is a free $R^{\bullet}(M_g)$-module of rank one.