We prove that every finite-dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

We extend the definition of Khovanov–Lee homology to links in connected sums of $S^1\times S^2$’s and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1\times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following 4-manifolds: $B^2\times S^2$, $S^1\times B^3$, ${\mathbb{CP}}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen’s invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are nonstandard.

We explain how non-Archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis, we work over a nonstandard model of the field of complex numbers, which is endowed at the same time with an Archimedean and a non-Archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of Archimedean and non-Archimedean forms which is compatible with integration.

We prove sharp ${\ell }^q{L}^p$ decoupling inequalities for $p,q\in [2,\mathrm{\infty })$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp–Lieb inequalities.