Let $\mathcal{T}$ be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the $\mathsf{\Pi }$ map provides a linear parametrization of the nonlinear space of admissible models $\mathsf{M}=(\mathsf{g},\mathsf{\Pi })$ on $\mathcal{T}$ in terms of the family of pararemainders used in the representation. We give an explicit description of the action of the most general class of renormalization schemes on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps. The BHZ renormalization scheme has that property.

We study the order of automorphism groups of cyclic covering fibrations of a ruled surface. Arakawa and later Chen studied it for hyperelliptic fibrations and gave the upper bound. The purpose of this paper is to pursue the analog for cyclic covering fibrations.

The main theorem is a classification of smooth actions of $SL(n,\mathbf{R})$, $n\ge 3$, or connected groups locally isomorphic to it, on closed n-manifolds, extending a theorem of Uchida. We also construct new exotic actions of $SL(n,\mathbf{Z})$ on the n-torus and connected sums of n-tori, and we formulate a conjectural classification of actions of lattices in $SL(n,\mathbf{R})$ on closed n-manifolds. We prove some related results about invariant rigid geometric structures for $SL(n,\mathbf{R})$-actions.

We develop the basic theory of Maurer–Cartan simplicial sets associated to (shifted complete) $L_{\mathrm{\infty }}$-algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the $L_{\mathrm{\infty }}$-algebra is concentrated in nonnegative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite group.

Let R and S be semiregular rings and U a semidualizing $(R,S)$-bimodule. We show that the U-codominant dimensions of ${}_{R}U$ and $U_S$ are identical. As an application, we get that the U-codominant dimension of U is at least two if and only if the functor $U{\otimes }_S{Hom}_R(U,-)$ is right exact and if and only if the functor ${Hom}_R(U,U{\otimes }_S-)$ is left exact. We also get some new equivalent characterizations of (n-)Auslander algebras.

Employing a variant of Hörmander’s approach to Andreotti–Grauert’s theory, a comparison theorem is proved between some bundle-valued weighted $L^2$ $\overline{\partial }$-cohomology groups of a class of locally pseudoconvex bounded domains in complex manifolds. As a result, such domains will turn out to be domains of meromorphy if the manifold admits a Hermitian line bundle whose curvature form is positive on the boundary of the domain. Particularly, a domain in a compact complex algebraic surface admits a meromorphic function whose essential singularities are the points on the boundary if the complement of the domain is a complex curve of self-intersection zero.