Extending Whitney's extension theorem: nonlinear function spaces

Abstract is in English and FrenchWe consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains C, with non-smooth boundary, in possibly non-compact manifolds. Assuming C is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where C only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator. = Nous considérons une version du problème de l’extension de Whitney, globale et non linéaire, pour les fonctions lisses à valeurs dans des variétés et définies sur des domaines fermés C à bords non-lisses dans des variétés possiblement non compactes. Supposant que C est une sous-variété à bord anguleux, ou qu’elle est compacte et localement convexe à bords non-lisses, nous montrons que l’opérateur de restriction, à partir des fonctions définies partout, est une submersion de variétés localement convexes, et donc possède des scindages linéaires locaux sur les cartes. Nous considérons à cet effet l’opérateur de restriction correspondant pour les espaces localement convexes de sections de fibrés vectoriels à support compact, permettant aussi de tariter le cas plus général où C n’a que des restrictions légères sur les cusps vers l’intérieur et l’extérieur, et montrons l’existence d’un opérateur de prolongement.David Michael Roberts and Alexander Schmedin

Biomarkers of disease differentiation: HCV recurrence versus acute cellular rejection

The wound-healing process induced by chronic hepatitis C virus (HCV) infection triggers liver damage characterized by fibrosis development and finally cirrhosis. Liver Transplantation (LT) is the optimal surgical treatment for HCV-cirrhotic patients at end-stage liver disease. However, acute cellular rejection (ACR) and HCV recurrence disease represent two devastating complications post-LT. The accurate differential diagnosis between both conditions is critical for treatment choice, and similar histological features represent a challenge for pathologists. Moreover, the HCV recurrence disease severity is highly variable post-LT. HCV recurrence disease progression is characterized by an accelerated fibrogenesis process, and almost 30% of those patients develop cirrhosis at 5-years of follow-up. Whole-genome gene expression (WGE) analyses through well-defined oligonucleotide microarray platforms represent a powerful tool for the molecular characterization of biological process. In the present manuscript, the utility of microarray technology is applied for the ACR and HCV-recurrence biological characterization in post-LT liver biopsy samples. Moreover, WGE analysis was performed to identify predictive biomarkers of HCV recurrence severity in formalin-fixed paraffin-embedded liver biopsies prospectively collected