On the Pierce-Birkhoff Conjecture

International audienceThis paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points$\alpha,\beta\in\sper\ A$and their separating ideal$$; we refer to this statement as the Local Pierce-Birkhoff conjecture at$\alpha,\beta$. In this paper, for each pair$(\alpha,\beta)$with$ht()=\dim A$, we define a natural number, called complexity of$(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points$\alpha,\beta$is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when$ht()$is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when$ht()$less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to$ht()=3$, the pair$(\alpha,\beta)$is of complexity 1 and$A$ is excellent with residue field the field of real numbers

Shifted Symplectic Structures

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks. We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures. Our main existence theorem states that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic structure, the derived mapping stack $\textbf{Map}(X,F)$ is equipped with a canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a Calabi-Yau condition in dimension $d$. These two results imply the existence of many examples of derived moduli stacks equipped with $n$-shifted symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We also show that Lagrangian intersections carry canonical (-1)-shifted symplectic structures.Comment: 52 pages. To appear in Publ. Math. IHE

On the structure of the graded algebra associated to a valuation

International audienceThe main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra gr v (R) of a subring (R, m) of a valuation ring Ov, for which Kv := Ov/mv = R/m, is isomorphic to Kv t v(R) , where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist

Algébrisation des variétés analytiques complexes et catégories dérivées.

International audienceWe prove that a compact complex analytic variety is algebraizable if and only if its bounded derived dg-category of coherent sheaves is saturated

HDAC1/2-dependent P0 expression maintains paranodal and nodal integrity independently of myelin stability through interactions with neurofascins

The pathogenesis of peripheral neuropathies in adults is linked to maintenance mechanisms that are not well understood. Here, we elucidate a novel critical maintenance mechanism for Schwann cell (SC)-axon interaction. Using mouse genetics, ablation of the transcriptional regulators histone deacetylases 1 and 2 (HDAC1/2) in adult SCs severely affected paranodal and nodal integrity and led to demyelination/remyelination. Expression levels of the HDAC1/2 target gene myelin protein zero (P0) were reduced by half, accompanied by altered localization and stability of neurofascin (NFasc)155, NFasc186, and loss of Caspr and septate-like junctions. We identify P0 as a novel binding partner of NFasc155 and NFasc186, both in vivo and by in vitro adhesion assay. Furthermore, we demonstrate that HDAC1/2-dependent P0 expression is crucial for the maintenance of paranodal/nodal integrity and axonal function through interaction of P0 with neurofascins. In addition, we show that the latter mechanism is impaired by some P0 mutations that lead to late onset Charcot-Marie-Tooth disease

Data from: HDAC1/2-dependent P0 expression maintains paranodal and nodal integrity independently of myelin stability through interactions with neurofascins

The pathogenesis of peripheral neuropathies in adults is linked to maintenance mechanisms that are not well understood. Here, we elucidate a novel critical maintenance mechanism for Schwann cell (SC)–axon interaction. Using mouse genetics, ablation of the transcriptional regulators histone deacetylases 1 and 2 (HDAC1/2) in adult SCs severely affected paranodal and nodal integrity and led to demyelination/remyelination. Expression levels of the HDAC1/2 target gene myelin protein zero (P0) were reduced by half, accompanied by altered localization and stability of neurofascin (NFasc)155, NFasc186, and loss of Caspr and septate-like junctions. We identify P0 as a novel binding partner of NFasc155 and NFasc186, both in vivo and by in vitro adhesion assay. Furthermore, we demonstrate that HDAC1/2-dependent P0 expression is crucial for the maintenance of paranodal/nodal integrity and axonal function through interaction of P0 with neurofascins. In addition, we show that the latter mechanism is impaired by some P0 mutations that lead to late onset Charcot-Marie-Tooth disease

Underlying data main and supplementary figures

2 Excel files with underlying data of Figs. 1CEFGH, 2AD, 3AGH, 4AB, 5BEGI, 6F, 7F, 9BE, S1ABC, S2BCD, S3ABCDE, S5, S6A, S7BC, S17BE, S18, and 2 video for Fig 1

Detached paranodal loops and wider nodes in dKO.

<p>Electron micrographs of ultrathin longitudinal control and dKO sciatic nerve sections at 8 wk post-tamoxifen showing in (A) paranodal loops attached to the axolemma and septate-like junctions (arrows) in control nerves, and detached paranodal loops devoid of septate-like junctions (arrows) in dKO nerves. In some dKO nodes, microvilli (highlighted in blue, image on the right) invaded the space between paranodal loops and the axolemma. Images on the right are magnifications of white boxes depicted on the left images. The graph representing the percentage of paranodes with detached loops in control and dKO demonstrates frequent occurrence of these defects in dKO sciatic nerves. Three animals per genotype were used, 11 to 38 paranodes were counted per animal, and 56 to 72 were counted per genotype. In (B), electron micrographs represent nodes of control (Ctr in the graph) and dKO nerves, and the quantification of nodal widths in the graph shows significant widening of the nodal region in dKO sciatic nerves. Three animals per genotype were used for quantification. The average width of 7 to 17 nodes of Ranvier was calculated per animal (<i>n</i> = 3), a total of 32 to 42 nodes were measured per genotype. Scale bars = 1 μm. In (A), error bar = SEM. In (B), the graph is a box plot where the lower box (Median − Quartile 1) and the upper box (Quartile 3 − Median) are separated by the Median value and flanked by top and bottom Whiskers. <i>P</i>-values (unpaired two-tailed Student's <i>t</i> test): *** = <i>p</i> < 0.001, <i>n</i> = 3.</p

The four P0 mutations D6Y, D32G, H52Y and S49L result in three different binding profiles to neurofascins: preserved (S49L), impaired binding to NFasc155 (D32G), impaired binding to both NFasc (D6Y and H52Y), while binding to P0 is maintained for all mutants.

<p>Adhesion assay in HEK293T cells. Confocal images of P0-Fc, P0-D6Y-Fc, P0-D32G-Fc, P0-H52Y-Fc, P0-S49L-Fc or control-Fc (Neg-Fc) particles (green) and neurofascins or Myc (red) coimmunofluorescence in HEK293T cells expressing P0-myc (A), NFasc155 (B) or NFasc186 (C), indicated by arrows. Overlays appear yellow. Nuclei are labeled in blue with DAPI. Single optical sections are shown. At least three independent experiments were analyzed for each panel and representative pictures are shown.</p