Functoriality and duality in Morse-Conley-Floer homology

In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known~\cite{abbondandoloschwarz, aizenbudzapolski,audindamian, kronheimermrowka, schwarz}, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolating map and flow map allows the results to generalize to local Morse homology and Morse-Conley-Floer homology. We prove Poincar\'e type duality statements for local Morse homology and Morse-Conley-Floer homology.Comment: To appear in the Journal of Fixed Point theory and its Application

Morse-Conley-Floer Homology

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology

Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation

Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on appropriate categories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups. Sheaves are constructed from such functors, which encode data about the continuation of structure as system parameters vary. Similarly, morphisms for the sheaves in question arise from natural transformations. This framework is applied to a variety of lattice algebras and ring structures associated to dynamical systems, whose algebraic properties carry over to their respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invariants which contain information about bifurcations of the parametrized systems

Suppression of planar cell polarity signaling and migration in glioblastoma by Nrdp1-mediated Dvl polyubiquitination.

The lethality of the aggressive brain tumor glioblastoma multiforme (GBM) results in part from its strong propensity to invade surrounding normal brain tissue. Although oncogenic drivers such as epidermal growth factor receptor activation and Phosphatase and Tensin homolog inactivation are thought to promote the motility and invasiveness of GBM cells via phosphatidylinostitol 3-kinase activation, other unexplored mechanisms may also contribute to malignancy. Here we demonstrate that several components of the planar cell polarity (PCP) arm of non-canonical Wnt signaling including VANGL1, VANGL2 and FZD7 are transcriptionally upregulated in glioma and correlate with poorer patient outcome. Knockdown of the core PCP pathway component VANGL1 suppresses the motility of GBM cell lines, pointing to an important mechanistic role for this pathway in glioblastoma malignancy. We further observe that restoration of Nrdp1, a RING finger type E3 ubiquitin ligase whose suppression in GBM also correlates with poor prognosis, reduces GBM cell migration and invasiveness by suppressing PCP signaling. Our observations indicate that Nrdp1 physically interacts with the Vangl1 and Vangl2 proteins to mediate the K63-linked polyubiquitination of the Dishevelled, Egl-10 and Pleckstrin (DEP) domain of the Wnt pathway protein Dishevelled (Dvl). Ubiquitination hinders Dvl binding to phosphatidic acid, an interaction necessary for efficient Dvl recruitment to the plasma membrane upon Wnt stimulation of Fzd receptor and for the propagation of downstream signals. We conclude that the PCP pathway contributes significantly to the motility and hence the invasiveness of GBM cells, and that Nrdp1 acts as a negative regulator of PCP signaling by inhibiting Dvl through a novel polyubiquitination mechanism. We propose that the upregulation of core PCP components, together with the loss of the key negative regulator Nrdp1, act coordinately to promote GBM invasiveness and malignancy

Non-filamentary (VMCO) memory : a two- and three-dimensional study on switching and failure modes

In this work, for the first time, a set of two-and three-dimensional (3D) analysis techniques are combined to clarify the nature of resistive switching (RS) in state-of-the-art TiO2-based vacancy modulated conductive oxide (VMCO) memory. (1) A non-filamentary switching mechanism is observed. (2) The role of oxygen incorporation and motion in the TiO2 is demonstrated. (3) The oxygen profile inside scaled cells is measured and a RS-model based on the modulation of oxygen inside the stack is proposed. In addition, we perform the tomographic analysis of fully-fabricated devices with Scalpel SPM, thus probing in 3D the entire stack and the contribution of TiO2 grain boundaries (GBs) to the switching operations. Finally, devices failed by breakdown (BD) during cycling are characterized, identifying the formation of parasitic filaments as root-cause of the failure