Dynamic asymptotic dimension for actions of virtually cyclic groups

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$ -algebras arising from minimal free $\mathbb {Z}$ -actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

On manifolds with infinitely many fillable contact structures

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak’s Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.

The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains

The Stokes resolvent problem $$\lambda u - \Delta u + \nabla \phi = f$$ λ u - Δ u + ∇ ϕ = f with $${\text {div}}(u) = 0$$ div ( u ) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of $$\mathrm {L}^2_{\sigma } (\Omega ) \ni f \mapsto \phi \in \mathrm {L}^2 (\Omega )$$ L σ 2 ( Ω ) ∋ f ↦ ϕ ∈ L 2 ( Ω ) decays like $$|\lambda |^{- 1 / 2}$$ | λ | - 1 / 2 which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like $$|\lambda |^{- \alpha }$$ | λ | - α for $$0 \le \alpha \le 1 / 4$$ 0 ≤ α ≤ 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain $$\Omega $$ Ω is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side $$f \in \mathrm {L}^2 (\Omega ; {\mathbb {C}}^d)$$ f ∈ L 2 ( Ω ; C d ) admit $$\mathrm {H}^2$$ H 2 -regularity and further prove localized $$\mathrm {H}^2$$ H 2 -estimates for the Stokes resolvent problem. By a generalized version of Shen’s $$\mathrm {L}^p$$ L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in $$\mathrm {L}^p (\Omega ; {\mathbb {C}}^d)$$ L p ( Ω ; C d ) for $$2d / (d + 2)< p < 2d / (d - 2)$$ 2 d / ( d + 2 ) < p < 2 d / ( d - 2 ) (with $$1< p < \infty $$ 1 < p < ∞ if $$d = 2$$ d = 2 ). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.

A Random Matrix Approach to Absorption in Free Products

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if ${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$, then ${\mathcal{P}}$ absorbs any subalgebra of ${\mathcal{M}}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra ${\mathcal{P}} \leq{\mathcal{M}}$ to have this absorption property, it suffices for ${\mathcal{M}}$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto ${\mathcal{P}}$.

Some properties of left 2α-Engel subgroups

In 1904, Schur proved his famous result which says that if the central factor group of a given group is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In the present paper, for a given automorphism [Formula: see text] of the group [Formula: see text], we introduce the concept of left [Formula: see text]-Engel, [Formula: see text], and [Formula: see text]-Engel commutator, [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

W1,p versus C1: The nonsmooth case involving critical growth

In this paper, we study a class of generalized and not necessarily differentiable functionals of the form [Formula: see text] with functions [Formula: see text], [Formula: see text] that are only locally Lipschitz in the second argument and involving critical growth for the elements of their generalized gradients [Formula: see text] even on the boundary [Formula: see text]. We generalize the famous result of Brezis and Nirenberg [[Formula: see text] versus [Formula: see text] local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317(5) (1993) 465–472] to a more general class of functionals and extend all the other generalizations of this result which has been published in the last decades.

A bound on the rate of convergence in the central limit theorem for renewal processes under second moment conditions

A famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. We provide an explicit non-uniform bound for the renewal central limit theorem based on Stein’s method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry–Esseen theorem under second moment conditions.

A Fox–Milnor theorem for knots in a thickened surface

A knot in a thickened surface [Formula: see text] is a smooth embedding [Formula: see text], where [Formula: see text] is a closed, connected, orientable surface. There is a bijective correspondence between knots in [Formula: see text] and knots in [Formula: see text], so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox–Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox–Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.

Some results of left and right α-commutators of groups

In [Formula: see text], Schur proved his famous result which says that if the central factor group of a given group [Formula: see text] is finite, then so is its derived subgroup. In [Formula: see text], Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In this paper, we introduce the concept of left and right [Formula: see text]-commutator, [Formula: see text], and [Formula: see text], where [Formula: see text] is an automorphism of the group [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.