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Thisvolume grew out of two Simons Symposia on "Nonarchimedean and tropicalgeometry" which took place on the island of St. John in April 2013 and inPuerto Rico in February 2015. Each meeting gathered a small group of expertsworking near the interface between tropical geometry and nonarchimedeananalytic spaces for a series of inspiring and provocative lectures on cuttingedge research, interspersed with lively discussions and collaborative work insmall groups. The articles collected here, which include high-level surveys aswell as original research, mirror the main themes of the two Symposia.
Topicscovered in this volume include:
- Differential forms and currents, andsolutions of Monge-Ampere type differential equations on Berkovich spaces andtheir skeletons;
- The homotopy types of nonarchimedean analytifications;
- The existence of "faithful tropicalizations" which encode the topology andgeometry of analytifications;
Relations between nonarchimedean analyticspaces and algebraic geometry, including logarithmic schemes, birationalgeometry, and the geometry of algebraic curves;- Extended notions oftropical varieties which relate to Huber's theory of adic spaces analogously tothe way that usual tropical varieties relate to Berkovich spaces; and
- Relationsbetween nonarchimedean geometry and combinatorics, including deep andfascinating connections between matroid theory, tropical geometry, and Hodgetheory.
List of contents
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].
About the author
Matthew BakerDepartment of MathematicsGeorgia Institute of TechnologyAtlanta, Georgia
mbaker@math.gatech.eduSam Payne
Department of Mathematics
Yale University
New Haven, Connecticut
sam.payne@yale.edu
Summary
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.
