Nonarchimedean and Tropical Geometry

This volume grew outof two Simons Symposia on Nonarchimedean and tropical geometrywhich took place on the island of St. John in April 2013 and in Puerto Rico inFebruary 2015. Each meeting gathered a small group of experts working near theinterface between tropical geometry and nonarchimedean analytic spaces for aseries of inspiring and provocative lectures on cutting edge research,interspersed with lively discussions and collaborative work in small groups.The articles collected here, which include high-level surveys as well asoriginal research, mirror the main themes of the two Symposia.Topics covered inthis volume include:  Differential forms and currents, and solutions of Monge-Ampère type differential equations on Berkovich spaces and their skeletons;  The homotopy types of nonarchimedean analytifications; The existence of faithful tropicalizations which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and  Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.