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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.

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Preface.- Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros) [W. Gubler].- Convergence Polygons for Connections on Nonarchimedean Curves [K.S. Kedlaya].- About Hrushovski and Loeser´s work on the Homotopy Type of Berkovich Spaces [A. Ducros].- Excluded Homeomorphism Types for Dual Complexes of Surfaces [D. Cartwright].- Analytification and Tropicalization over Non-Archimedean Fields [A. Werner].- Berkovich Skeleta and Birational Geometry [J. Nicaise].- Metrization of Differential Pluriforms on Berkovich Analytic Spaces [M. Temkin].- Skeletons and Fans of Logarithmic Structures [D. Abramovich, Q. Chen, S. Marcus, M. Ulirsch, and J. Wise].- Introduction to Adic Tropicalization [T. Foster].- Degeneration of Linear Series from the Tropical Point of View and Applications [M. Baker and D. Jensen].- Matroid Theory for Algebraic Geometries [E. Katz].

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