Nonarchimedean and Tropical Geometry / Simons Symposia (PDF)

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.