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Preface; 1 Introduction; 2 Contents; Contents; List of Contributors; Forms and Currents on the Analytification of an Algebraic Variety (After Chambert-Loir and Ducros); 1 Introduction; 2 Superforms and Supercurrents on Rr; 3 Superforms on Polyhedral Complexes; 4 Moment Maps and Tropical Charts; 5 Differential Forms on Algebraic Varieties; 6 Currents on Algebraic Varieties; 7 Generalizations to Analytic Spaces; References; The Non-Archimedean Monge-Ampère Equation; 1 Introduction; 2 Metrics on Lines Bundles; 3 The Monge-Ampère Operator; 4 The Complex Monge-Ampère Equation
5 The Non-Archimedean Monge-Ampère Equation6 A Variational Approach; 7 Singular Semipositive Metrics; 8 Energy; 9 Envelopes, Differentiability, and Orthogonality; 10 Curves; 11 Toric Varieties; 12 Outlook; References; Convergence Polygons for Connections on Nonarchimedean Curves; 1 Newton Polygons; 2 PL Structures on Berkovich Curves; 3 Convergence Polygons: Projective Line; 4 A Gallery of Examples; 5 Convergence Polygons: General Curves; 6 Derivatives of Convergence Polygons; 7 Subharmonicity and Index; 8 Ramification of Finite Morphisms; 9 Artin-Hasse Exponentials and Witt Vectors
10 Kummer-Artin-Schreier-Witt Theory11 Automorphisms of a Formal Disc; Appendix 1: Convexity; Appendix 2: Thematic Bibliography; References; About Hrushovski and Loeser's Work on the Homotopy Type of Berkovich Spaces; 1 Introduction; 2 Model Theory of Valued Fields: Basic Definitions; 3 Hrushovski and Loeser's Fundamental Construction; 4 Homotopy Type of and Links with Berkovich Spaces; 5 An Application of the Definability of for C a Curve; References; Excluded Homeomorphism Types for Dual Complexes of Surfaces; 1 Introduction; 2 Tropical Complexes and Tropical Surfaces; 3 Degenerations
4 Proof of the Main TheoremsReferences; Analytification and Tropicalization Over Non-archimedean Fields; 1 Introduction; 2 Berkovich Spaces and Tropicalizations; 2.1 Notation and Conventions; 2.2 Berkovich Spaces; 2.3 Tropicalization; 3 The Case of Curves; 4 Tropical Grassmannians; 4.1 The Setting; 4.2 A Section of the Tropicalization Map; 4.3 Sketch of Proof in the Dense Torus Orbit; 5 Skeleta of Semistable Pairs; 5.1 Integral Affine Structures; 5.2 Semistable Pairs; 5.3 Skeleta; 6 Functions on the Skeleton; 7 Faithful Tropicalizations; 7.1 Finding a Faithful Tropicalization for a Skeleton
7.2 Finding a Copy of the Tropicalization Inside the Analytic SpaceReferences; Berkovich Skeleta and Birational Geometry; 1 Introduction; 2 The Berkovich Skeleton of an sncd-Model; 2.1 Birational Points; 2.2 Models; 2.3 Divisorial and Monomial Points; 2.4 The Berkovich Skeleton; 2.5 The Deformation Retraction in a Basic Example; 3 Weight Functions and the Kontsevich-Soibelman Skeleton; 3.1 The Work of Kontsevich and Soibelman; 3.2 Log Discrepancies in Birational Geometry; 3.3 Definition of the Kontsevich-Soibelman Skeleton; 3.4 Definition and Properties of the Weight Function
5 The Non-Archimedean Monge-Ampère Equation6 A Variational Approach; 7 Singular Semipositive Metrics; 8 Energy; 9 Envelopes, Differentiability, and Orthogonality; 10 Curves; 11 Toric Varieties; 12 Outlook; References; Convergence Polygons for Connections on Nonarchimedean Curves; 1 Newton Polygons; 2 PL Structures on Berkovich Curves; 3 Convergence Polygons: Projective Line; 4 A Gallery of Examples; 5 Convergence Polygons: General Curves; 6 Derivatives of Convergence Polygons; 7 Subharmonicity and Index; 8 Ramification of Finite Morphisms; 9 Artin-Hasse Exponentials and Witt Vectors
10 Kummer-Artin-Schreier-Witt Theory11 Automorphisms of a Formal Disc; Appendix 1: Convexity; Appendix 2: Thematic Bibliography; References; About Hrushovski and Loeser's Work on the Homotopy Type of Berkovich Spaces; 1 Introduction; 2 Model Theory of Valued Fields: Basic Definitions; 3 Hrushovski and Loeser's Fundamental Construction; 4 Homotopy Type of and Links with Berkovich Spaces; 5 An Application of the Definability of for C a Curve; References; Excluded Homeomorphism Types for Dual Complexes of Surfaces; 1 Introduction; 2 Tropical Complexes and Tropical Surfaces; 3 Degenerations
4 Proof of the Main TheoremsReferences; Analytification and Tropicalization Over Non-archimedean Fields; 1 Introduction; 2 Berkovich Spaces and Tropicalizations; 2.1 Notation and Conventions; 2.2 Berkovich Spaces; 2.3 Tropicalization; 3 The Case of Curves; 4 Tropical Grassmannians; 4.1 The Setting; 4.2 A Section of the Tropicalization Map; 4.3 Sketch of Proof in the Dense Torus Orbit; 5 Skeleta of Semistable Pairs; 5.1 Integral Affine Structures; 5.2 Semistable Pairs; 5.3 Skeleta; 6 Functions on the Skeleton; 7 Faithful Tropicalizations; 7.1 Finding a Faithful Tropicalization for a Skeleton
7.2 Finding a Copy of the Tropicalization Inside the Analytic SpaceReferences; Berkovich Skeleta and Birational Geometry; 1 Introduction; 2 The Berkovich Skeleton of an sncd-Model; 2.1 Birational Points; 2.2 Models; 2.3 Divisorial and Monomial Points; 2.4 The Berkovich Skeleton; 2.5 The Deformation Retraction in a Basic Example; 3 Weight Functions and the Kontsevich-Soibelman Skeleton; 3.1 The Work of Kontsevich and Soibelman; 3.2 Log Discrepancies in Birational Geometry; 3.3 Definition of the Kontsevich-Soibelman Skeleton; 3.4 Definition and Properties of the Weight Function