Introduction to Tropical Geometry

Preface
Chapter 1. Tropical Islands
1.1. Arithmetic
1.2. Dynamic Programming
1.3. Plane Curves
1.4. Amoebas and their Tentacles
1.5. Implicitization
1.6. Group Theory
1.7. Curve Counting
1.8. Compactifications
1.9. Exercises
Chapter 2. Building Blocks
2.1. Fields
2.2. Algebraic Varieties
2.3. Polyhedral Geometry
2.4. Gröbner Bases
2.5. Gröbner Complexes
2.6. Tropical Bases
2.7. Exercises
Chapter 3. Tropical Varieties
3.1. Hypersurfaces
3.2. The Fundamental Theorem
3.3. The Structure Theorem
3.4. Multiplicities and Balancing
3.5. Connectivity and Fans
3.6. Stable Intersection
3.7. Exercises
Chapter 4. Tropical Rain Forest
4.1. Hyperplane Arrangements
4.2. Matroids
4.3. Grassmannians
4.4. Linear Spaces
4.5. Surfaces
4.6. Complete Intersections
4.7. Exercises
Chapter 5. Tropical Garden
5.1. Eigenvalues and Eigenvectors
5.2. Tropical Convexity
5.3. The Rank of a Matrix
5.4. Arrangements of Trees
5.5. Monomials in Linear Forms
5.6. Exercises
Chapter 6. Toric Connections
6.1. Toric Background
6.2. Tropicalizing Toric Varieties
6.3. Orbits
6.4. Tropical Compactifications
6.5. Geometric Tropicalization
6.6. Degenerations
6.7. Intersection Theory
6.8. Exercises
Bibliography
Index