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–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.