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 Matemàtica (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 Poincaré 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.
