We show that the Taylor coefficients of the series $q(z)=z\exp (G(z)/F(z))$ are integers, where $F(z)$ and $G(z)+\log (z)F(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z^{-1}q(z){)}^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau and by Zudilin. In particular, we prove the general “integrality” conjecture of Zudilin about these mirror maps

We exhibit cocycles representing certain classes in the cohomology of the algebraic group ${\mathrm{GL}}_n$ with coefficients in the representation ${\Gamma }^{*}({\mathfrak{gl}}_n^{(1)})$. These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras (see [18])

Let $G$ be a reductive linear algebraic group over a field $k$. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. Invariant theory states that the ring of invariants $A^G=H^0(G,A)$ is finitely generated. We show that in fact the full cohomology ring $H^{*}(G,A)$ is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed in [22]. We also continue the study of bifunctor cohomology of ${\Gamma }^{*}({\mathfrak{gl}}^{(1)})$

Let M be a hyperbolic $3$-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, answering a question of Minsky. This is proved via a filling theorem, which states that as one performs higher and higher Dehn fillings, the skinning maps converge uniformly on all of Teichmüller space.

We also exhibit manifolds with totally geodesic boundaries whose skinning maps have diameter tending to infinity, as well as manifolds whose skinning maps have diameter tending to zero (the latter are due to Bromberg and Kent).

In the final section, we give a proof of Thurston's bounded image theorem