On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio

No-splitting property and boundaries of random groups

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.Comment: 20 page

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.Comment: 11 pages, title changed, a part of contents remove

An algorithm to identify automorphisms which arise from self-induced interval exchange transformations

We give an algorithm to determine if the dynamical system generated by a positive automorphism of the free group can also be generated by a self-induced interval exchange transformation. The algorithm effectively yields the interval exchange transformation in case of success.Comment: 26 pages, 8 figures. v2: the article has been reorganized to make for a more linear read. A few paragraphs have been added for clarit

Plate tectonics drive tropical reef biodiversity dynamics

The Cretaceous breakup of Gondwana strongly modified the global distribution of shallow tropical seas reshaping the geographic configuration of marine basins. However, the links between tropical reef availability, plate tectonic processes and marine biodiversity distribution patterns are still unknown. Here, we show that a spatial diversification model constrained by absolute plate motions for the past 140 million years predicts the emergence and movement of diversity hotspots on tropical reefs. The spatial dynamics of tropical reefs explains marine fauna diversification in the Tethyan Ocean during the Cretaceous and early Cenozoic, and identifies an eastward movement of ancestral marine lineages towards the Indo-Australian Archipelago in the Miocene. A mechanistic model based only on habitat-driven diversification and dispersal yields realistic predictions of current biodiversity patterns for both corals and fishes. As in terrestrial systems, we demonstrate that plate tectonics played a major role in driving tropical marine shallow reef biodiversity dynamics

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

To respond or not to respond - a personal perspective of intestinal tolerance

For many years, the intestine was one of the poor relations of the immunology world, being a realm inhabited mostly by specialists and those interested in unusual phenomena. However, this has changed dramatically in recent years with the realization of how important the microbiota is in shaping immune function throughout the body, and almost every major immunology institution now includes the intestine as an area of interest. One of the most important aspects of the intestinal immune system is how it discriminates carefully between harmless and harmful antigens, in particular, its ability to generate active tolerance to materials such as commensal bacteria and food proteins. This phenomenon has been recognized for more than 100 years, and it is essential for preventing inflammatory disease in the intestine, but its basis remains enigmatic. Here, I discuss the progress that has been made in understanding oral tolerance during my 40 years in the field and highlight the topics that will be the focus of future research

Centriole splitting caused by loss of the centrosomal linker protein C-NAP1 reduces centriolar satellite density and impedes centrosome amplification

Duplication of the centrosomes is a tightly regulated process. Abnormal centrosome numbers can impair cell division and cause changes in how cells migrate. Duplicated centrosomes are held together by a proteinaceous linker made up of rootletin filaments anchored to the centrioles by C-NAP1. This linker is removed in a NEK2A kinase-dependent manner as mitosis begins. To explore C-NAP1 activities in regulating centrosome activities, we used genome editing to ablate it. C-NAP1–null cells were viable and had an increased frequency of premature centriole separation, accompanied by reduced density of the centriolar satellites, with reexpression of C-NAP1 rescuing both phenotypes. We found that the primary cilium, a signaling structure that arises from the mother centriole docked to the cell membrane, was intact in the absence of C-NAP1, although components of the ciliary rootlet were aberrantly localized away from the base of the cilium. C-NAP1–deficient cells were capable of signaling through the cilium, as determined by gene expression analysis after fluid flow–induced shear stress and the relocalization of components of the Hedgehog pathway. Centrosome amplification induced by DNA damage or by PLK4 or CDK2 overexpression was markedly reduced in the absence of C-NAP1. We conclude that centriole splitting reduces the local density of key centriolar precursors to impede overduplication

Bioreactor analyses of tissue ingrowth, ongrowth and remodelling around implants: an alternative to live animal testing

Introduction: Preclinical assessment of bone remodelling onto, into or around novel implant technologies is underpinned by a large live animal testing burden. The aim of this study was to explore whether a lab-based bioreactor model could provide similar insight. Method: Twelve ex vivo trabecular bone cylinders were extracted from porcine femora and were implanted with additively manufactured stochastic porous titanium implants. Half were cultured dynamically, in a bioreactor with continuous fluid flow and daily cyclic loading, and half in static well plates. Tissue ongrowth, ingrowth and remodelling around the implants were evaluated with imaging and mechanical testing. Results: For both culture conditions, scanning electron microscopy (SEM) revealed bone ongrowth; widefield, backscatter SEM, micro computed tomography scanning, and histology revealed mineralisation inside the implant pores; and histology revealed woven bone formation and bone resorption around the implant. The imaging evidence of this tissue ongrowth, ingrowth and remodelling around the implant was greater for the dynamically cultured samples, and the mechanical testing revealed that the dynamically cultured samples had approximately three times greater push-through fixation strength (p < 0.05). Discussion: Ex vivo bone models enable the analysis of tissue remodelling onto, into and around porous implants in the lab. While static culture conditions exhibited some characteristics of bony adaptation to implantation, simulating physiological conditions with a bioreactor led to an accelerated response

Stabilizers of R\mathbb R-trees with free isometric actions of FNF_N

We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(\phi)$ is the forward limit tree of an atoroidal iwip element $\phi\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of $Out(F_N)$ is due to Bestvina, Feighn and Handel.Comment: corrected the proof of Proposition 4.1, plus several minor fixes and updates; to appear in Journal of Group Theor