We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if $\unicode[STIX]{x1D6E4}$ is a nonamenable group and $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann. 294 (1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France 133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, that is, where the interarrival law of the renewal process is given by $\text{K}(n)=n^{-3/2}\unicode[STIX]{x1D719}(n)$ where $\unicode[STIX]{x1D719}$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FD}\rightarrow 0}\unicode[STIX]{x1D6FD}^{2}\log h_{c}(\unicode[STIX]{x1D6FD})=-\frac{\unicode[STIX]{x1D70B}}{2}.\end{eqnarray}$$

For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category ${\mathcal{O}}$ and Koszul duality.

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite) $W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

We consider a general fibre of given length in a generic projection of a variety. Under the assumption that the fibre is of local embedding dimension 2 or less, an assumption which can be checked in many cases, we prove that the fibre is reduced and its image on the projected variety is an ordinary multiple point.

In this article, we construct examples of $L$-indistinguishable overconvergent eigenforms for an inner form of $\text{SL}_{2}$.

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].