This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic $K$-theory.

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

The absolutely continuous spectrum of a very general class of differential operators of order $2n$ is determined, for operators whose coefficients satisfy conditions that combine smoothness and decay properties. The main methods are asymptotic integration and the analysis of the associated $M$-matrix. The form of the solutions precludes the absence of a singular continuous spectrum.

We determine when a metric measure space admits a uniformizing density which has Ahlfors regular volume growth. A characterization of uniformizing conformal densities in terms of doubling and a lifting procedure are the key ingredients in our presentation. We also furnish an application characterizing metric doubling measures.

We correct the statements of Theorem 1 and Corollary 2 of the above-mentioned paper and provide a new argument for Corollary 2. We also update some parts of the paper in the light of recent major developments in the theory of bihermitian structures, due to M. Gualtieri and N. Hitchin.

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

We investigate Riemann--Liouville processes $R_H$, with $H > 0$, and fractional Brownian motions $B_H$, for $0 < H < 1$, and study their small deviation properties in the spaces $L_q([0, 1], \mu)$. Of special interest here are thin (fractal) measures $\mu$, that is, those that are singular with respect to the Lebesgue measure. We describe the behavior of small deviation probabilities by numerical quantities of $\mu$, called mixed entropy numbers, characterizing size and regularity of the underlying measure. For the particularly interesting case of self-similar measures, the asymptotic behavior of the mixed entropy is evaluated explicitly. We also provide two-sided estimates for this quantity in the case of random measures generated by subordinators.

While the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods, the lower bound is verified by analytic tools concerning entropy and Kolmogorov numbers of Riemann--Liouville operators.

A family $\mathbf{A}_\alpha$ of differential operators depending on a real parameter $\alpha \ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $\sigma_{a.c.}$ of the operator $\mathbf{A}_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $\mathbf{A}_0$ is purely absolutely continuous and admits an explicit description. It turns out that for $\alpha < \sqrt 2$ one has $\sigma_{a.c.}(\mathbf{A}_\alpha) = \sigma_{a.c.}(\mathbf{A}_0)$, including the multiplicity. For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary Jacobi matrix which is related to the operator $\mathbf{A}_\alpha$. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.