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What is Divisors and Sandpiles about?

Cover 1 Title page 4 Contents 8 Preface 12 Part 1 . Divisors 16 Chapter 1. The dollar game 18 1.1. An initial game 18 1.2. Formal definitions 20 1.3. The Picard and Jacobian groups 25 Notes 27 Problems for Chapter 1 28 Chapter 2. The Laplacian 30 2.1. The discrete Laplacian 30 2.2. Configurations and the reduced Laplacian 35 2.3. Complete linear systems and convex polytopes 39 2.4. Structure of the Picard group 42 Notes 50 Problems for Chapter 2 51 Chapter 3. Algorithms for winning 54 3.1. Greed 54 3.2. q-reduced divisors 57 3.3. Superstable configurations 60 3.4. Dhar’s algorithm and efficient implementation 61 3.5. The Abel-Jacobi map 65 Notes 68 Problems for Chapter 3 69 Chapter 4. Acyclic orientations 72 4.1. Orientations and maximal unwinnables 72 4.2. Dhar’s algorithm revisited 74 Notes 78 Problems for Chapter 4 79 Chapter 5. Riemann-Roch 80 5.1. The rank function 80 5.2. Riemann-Roch for graphs 82 5.3. The analogy with Riemann surfaces 85 5.4. Alive divisors and stability 93 Notes 95 Problems for Chapter 5 97 Part 2 . Sandpiles 100 Chapter 6. The sandpile group 102 6.1. A first example 102 6.2. Directed graphs 107 6.3. Sandpile graphs 108 6.4. The reduced Laplacian 112 6.5.

Author

Scott Corry,David Perkinson

Language

EN