This
volume grew out of two Simons Symposia on "Nonarchimedean and tropical
geometry" which took place on the island of St. John in April 2013 and in
Puerto Rico in February 2015. Each meeting gathered a small group of experts
working near the interface between tropical geometry and nonarchimedean
analytic spaces for a series 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 as original research, mirror the main themes of the two Symposia.

Topics
covered in this volume include: 

• Differential forms and currents, and
  solutions of Monge-Ampere 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.