In this article, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the decomposition theorem, as well as for the family of local systems on each support. It generalizes Gabber’s result on the independence of $\ell$ of intersection cohomology to the relative case.

We study the minimum problem for the functional $${\int }_{\Omega }\left(\right|\nabla \mathbf{u}{|}^2+Q^2{\chi }_{\left\{\right|\mathbf{u}|>0\}})dx$$ with the constraint $u_i\ge 0$ for $i=1,\dots ,m$, where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain and $\mathbf{u}=(u_1,\dots ,u_m)\in H^1(\Omega ;{\mathbb{R}}^m)$. First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\left\{\right|u|>0\}$ is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega \cap \partial \left\{\right|u|>0\}$.

We establish a Cheeger–Müller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.

The Grothendieck–Katz $p$-curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$, for almost all primes $p$. We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.