In 1997 Springer published a second edition of the book The Algebraic Theory of Spinors by Claude Chevalley (cf. The algebraic theory of spinors, 1197), initially published in 1954 by Columbia University Press. On the Springer side, Catriona BYRNE was responsible for the project. (It was part of a more comprehensive endeavour supported by the CNRS to publish Claude Chevalley’s Complete Works, that also involved Michel Broué.) Because, in the interval between the two editions, so much had happened in Mathematics and in Physics involving spinors I was asked to provide a Postface to the second edition (cf. Bourguignon, Postface: spinors in 1995, 1997). This is one of the opportunities offered to me by Catriona over the years. At this time, spinors were indeed one of my main objects of study. I decided to entitle this contribution Spinors in 1995. Since then, the topic of spinors continued to attract a lot of attention. This is why, for the special volume of the Lecture Notes in Mathematics dedicated to Catriona, I decided to contribute a follow up to the Postface. It was therefore natural to entitle it Spinors in 2022. I hope this short article will convince you of the appropriateness of the initiative.

Letter to the Editor

In this chapter, we present the main families of infinite-dimensional spaces, namely Banach spaces and Hilbert spaces.

In this final chapter, we study the relation between the presence of symmetries and the existence of conserved quantities in a variational problem. This theme plays a very important role both conceptually and from a practical point of view, because of the simplifications it allows in calculations.

The object of the first three chapters is to provide the analytic framework for the development of Variational Calculus. We present a number of basic ideas of functional analysis in a language which is deliberately geometric. To avoid making these chapters too lengthy, the basic definitions of topology used in the study of metric spaces are collected in an appendix which should be regarded as a glossary rather than as a course on this subject. For a more systematic introduction to topology, the interested reader may refer to (Choquet 1984) or (Schwartz 1970).

— This is a survey of a collection of related results about the deformation properties of Einstein metrics on a certain class of spaces with stratified singular structure. The results in low dimensions are particularly clean, and are motivated by applications in hyperbolic and convex geometry. The three-dimensional setting is related to an old conjecture by Stoker about flexibility of convex hyperbolic polyhedra, and we report on a partial answer. We also review some of the analytic methods used to prove these results. Résumé (Flexibilité des métriques d'Einstein singulières). — Cet article constitue un compte-rendu d'une collection de résultats autour des propriétés de déformation des métriques d'Einstein sur une certaine classe d'espaces à structure singulière stratifiée. Les résultats en basse dimension sont particulièrement intéressants, et ils sont motivés par des applications en géométrie hyperbolique et convexe. La configuration 3-dimensionnelle est reliée à une vieille conjecture de Stoker sur la flexibilité des polyèdres convexes hyperboliques et nous proposons une réponse partielle. Nous examinons également certaines méthodes analytiques utilisées pour démontrer ces résultats.

Searchable abstracts of presentations at key conferences in endocrinology ISSN 1470-3947 (print) | ISSN 1479-6848 (online)

Endocrine disrupting chemicals (EDCs) are a rising concern for public health due to their ubiquitous presence as complex mixtures affecting development throughout generations. Our goal was to study the effect of a mixture of EDCs on female sexual development during 3 generations. Female rats (F0 generation) were orally exposed to a mixture of 14 anti-androgenic and estrogenic EDCs or corn oil for 2 weeks before and throughout gestation and until weaning. The mixture was composed of plasticizers (BPA, DBP, DEHP), fungicides/pesticides (Vinclozolin, Procymidon, Prochloraz, Epoxynazole, Linurone, p-p’-DDT), UV filters (4-MBC, OMC), Butyl Paraben and the analgesic Acetaminophen. The doses were in the micrograms/kg range in order to represent human exposure. Sexual development (vaginal opening, GnRH interpulse interval and estrous cyclicity) as well as maternal behavior were studied from F1 to F3 generations. At PND21 the mediobasal hypothalamus of the F1 and F3 were removed for gene expression analysis by RNAseq and RT-qPCR as well as for Chromatin Immunoprecipitation of histone modifications at regulatory regions of target genes. While F2 and F3 females showed delayed vaginal opening, decreased percentage of regular estrous cycles and decreased GnRH interpulse interval, no such changes were detected in F1 animals. These reproductive phenotypes were associated with alterations in both transcriptional and histone posttranslational modifications of hypothalamic genes involved in reproductive competence and behavior like kisspeptin (Kiss1), oxytocin (Oxt), estrogen (Esr1), glutamate (Grin2d), dopamine signaling (Th and Drd1) as well as glucocorticoid activity (Nr3c1 and Crh). Concomitant with a decrease in transcriptional activity, we have observed either a decrease of active histone marks (H3K4me3, H3K9ac) for Esr1 and Oxt promoter regions, an increase of repressive histone modifications (H3K27me3, H3K9me3) for Grin2D, Th and Nr3c1 promoter regions or both for the Kiss1 promoter. Up-regulated genes (Pomc, and CRH) showed decreased H3K9me3 and increased H3K9ac at their 5’regulatory regions. F1 females that were exposed in utero to the EDC mixture, showed a reduction in Th mRNA expression and decreased grooming/licking behavior while spending more time resting alone. These alterations on maternal behavior are known to cause transgenerational alterations of the development of the corticotropic and gonadotropic axis. Overall, our data shows that gestational and lactational exposure to an environmentally relevant EDC mixture transgenerationally affects sexual development throughout epigenetic reprogramming of the hypothalamus. Such effects could be mediated by alterations of maternal behavior caused by exposure of the first generation to the EDC mixture.

Background The regulatory management of chemicals and toxicants in the EU addresses hundreds of different chemicals and health hazards individually, one by one. An issue is that, so far, the possible interactions among chemicals or hazards are not considered as such. Another issue is the anticipated delay of several decades before effective protection of public health by regulatory decisions due to a time consuming process. Prenatal and early postnatal life is highly vulnerable to environmental health hazards with lifelong consequences, and a priority period for reduction of exposure. There are some initiatives regarding recommendations for pregnant women aiming at protection against one or another category of health hazard, however not validated by intervention studies. Hypothesis Here, we aim at strengthening the management of exposure to individual health hazards during pregnancy and lactation, with protective measures in a global strategy of Environmental Hygiene. We hypothesize that such a strategy could reduce both the individual effects of harmful agents in complex mixtures and the possible interactions among them. A panel of experts should develop and endorse implementable measures towards a protective behavior. Their application is meant to be preferably as a package of measures in order to maximize protection and minimize interactions in causing adverse effects. Testing our hypothesis requires biomonitoring studies and longitudinal evaluation of health endpoints in the offspring. Favorable effects would legitimate further action towards equal opportunity access to improved environmental health. Conclusion Environmental Hygiene is proposed as a global strategy aiming at effective protection of pregnant women, unborn children and infants against lifelong consequences of exposure to combinations of adverse lifestyle factors.

Endocrine disruption is commonly thought to be restricted to a direct endocrine mode of action i.e. the perturbation of the activation of a given type of hormonal receptor by its natural ligand. Consistent with the WHO definition of an endocrine disrupter, a key issue is the "altered function(s) of the endocrine system". Such altered functions can result from different chemical interactions, beyond agonistic or antagonistic effect at a given receptor. Based on neuroendocrine disruption by polychlorinated biphenyls and bisphenol A, this paper proposes different mechanistic paradigms that can result in adverse health effects. They are a consequence of altered endocrine function(s) secondary to chemical interaction with different steps in the physiological regulatory processes, thus accounting for a possibly indirect endocrine mode of action.

one of the world's leading differential geometers and a corresponding member of the French Academy of Sciences for half a century, passed away on October 15, 2016, at the age of eighty-nine.Marcel Berger's contributions to geometry were both broad and deep.The classification of Riemannian holonomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to algebraic geometry.His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the

Background Endocrine-disrupting chemicals (EDCs) contribute to disease and dysfunction and incur high associated costs (>1% of the gross domestic product [GDP] in the European Union). Exposure to EDCs varies widely between the USA and Europe because of differences in regulations and, therefore, we aimed to quantify disease burdens and related economic costs to allow comparison.Methods We used existing models for assessing epidemiological and toxicological studies to reach consensus on probabilities of causation for 15 exposure-response relations between substances and disorders. We used Monte Carlo methods to produce realistic probability ranges for costs across the exposure-response relation, taking into account uncertainties. Estimates were made based on population and costs in the USA in 2010. Costs for the European Union were converted to US$ ((sic)1=$1.33).Findings The disease costs of EDCs were much higher in the USA than in Europe ($340 billion [2.33% of GDP] vs $217 billion [1.28%]). The difference was driven mainly by intelligence quotient (IQ) points loss and intellectual disability due to polybrominated diphenyl ethers (11 million IQ points lost and 43 000 cases costing $266 billion in the USA vs 873 000 IQ points lost and 3290 cases costing $12.6 billion in the European Union). Accounting for probability of causation, in the European Union, organophosphate pesticides were the largest contributor to costs associated with EDC exposure ($121 billion), whereas in the USA costs due to pesticides were much lower ($42 billion).Interpretation EDC exposure in the USA contributes to disease and dysfunction, with annual costs taking up more than 2% of the GDP. Differences from the European Union suggest the need for improved screening for chemical disruption to endocrine systems and proactive prevention.

Our society is facing a public health challenge caused by the increasing presence of endocrine disrupting chemicals (EDCs). Bisphenol A (BPA) is a widespread EDC used in the manufacture of polyvinyl chloride (PVC) and epoxy resins. As our group have previously described, an early postnatal exposure to BPA disrupts sexual maturation andpubertal timing, however, itseffects on fertility after adult exposure have not yet been studied.