This volume contains extended abstracts outlining selected talks and other selected presentations given by participants of the workshop "Positivity and Valuations", held at the Centre de Recerca Matematica (CRM) in Barcelona from February 22nd to 26th, 2016. They include brief research articles reporting new results, descriptions of preliminary work or open problems, and the outcome of work in groups initiated during the workshop. The general subject is the application of valuation theory to positivity questions in algebraic geometry. The topics covered range from purely algebraic problems like finite generation of semigroups and algebras defined by valuations, and properties of the associated Poincare series, to more geometric questions like resolution of singularities and properties of Newton-Okounkov bodies, linked with non-archimedean geometry and tropical geometry. The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.

This book provides a self-contained exposition of the theory of plane Cremona maps, reviewing the classical theory. The book updates, correctly proves and generalises a number of classical results by allowing any configuration of singularities for the base points of the plane Cremona maps. It also presents some material which has only appeared in research papers and includes new, previously unpublished results. This book will be useful as a reference text for any researcher who is interested in the topic of plane birational maps.

This book comprises an overview of twelve months of intense activity of the research group Geometry, Topology, Algebra, and Applications (GEOMVAP) at the Universitat Politecnica de Catalunya (UPC). Namely, it contains extended abstracts of the group meeting in Cardona and of the international Workshop of Women in Geometry and Topology aligned with a series of workshops in the topic. As such, it includes a panoramic view of the main research interests of the group which focus on varieties and manifolds from the algebraic, topological and differential perspective with a view towards applications. The GEOMVAP group has a long tradition working on various interfaces of algebra, geometry and topology. In the last decade, the group has become active contributor in interdisciplinary science and it is now focused on both a theoretical point of view and the transversal applications to several disciplines including Robotics, Machine Learning, Phylogenetics, Physics and Celestial Mechanics. The increasing interdisciplinarity of modern research and the fact that the boundaries between different areas of mathematics are vanishing, with a constant transfer of problems and techniques between them, makes it difficult to progress without a multidisciplinary approach. GEOMVAP gathers together experts in Algebraic, Symplectic and Arithmetic Geometry to stimulate the interaction between them and to allow the study of each object from different points of view. The book aims at established researchers, as well as at PhD and postdoctoral students who want to learn more about the latest advances in pure and applied Geometry and Topology.