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Divisors and Sandpiles: An Introduction to Chip-Firing
 
Scott Corry Lawrence University, Appleton, WI
David Perkinson Reed College, Portland, OR
Divisors and Sandpiles
Softcover ISBN:  978-1-4704-4218-7
Product Code:  MBK/114
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-4861-5
Product Code:  MBK/114.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $60.00
Softcover ISBN:  978-1-4704-4218-7
eBook: ISBN:  978-1-4704-4861-5
Product Code:  MBK/114.B
List Price: $164.00 $126.50
MAA Member Price: $147.60 $113.85
AMS Member Price: $131.20 $101.20
Divisors and Sandpiles
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Divisors and Sandpiles: An Introduction to Chip-Firing
Scott Corry Lawrence University, Appleton, WI
David Perkinson Reed College, Portland, OR
Softcover ISBN:  978-1-4704-4218-7
Product Code:  MBK/114
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-4861-5
Product Code:  MBK/114.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $60.00
Softcover ISBN:  978-1-4704-4218-7
eBook ISBN:  978-1-4704-4861-5
Product Code:  MBK/114.B
List Price: $164.00 $126.50
MAA Member Price: $147.60 $113.85
AMS Member Price: $131.20 $101.20
  • Book Details
     
     
    2018; 325 pp
    MSC: Primary 05;

    Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces.

    Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book.

    Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, \(M\)-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.

    Readership

    Undergraduate and graduate students and researchers interested in games on graphs, Riemann surfaces, finite abelian groups, and Markov chains.

  • Table of Contents
     
     
    • Divisors
    • The dollar game
    • The Laplacian
    • Algorithms for winning
    • Acyclic orientations
    • Riemann-Roch
    • Sandpiles
    • The sandpile group
    • Burning and duality
    • Threshold density
    • Topics
    • Trees
    • Harmonic morphisms
    • Divisors on complete graphs
    • More about sandpiles
    • Cycles and cuts
    • Matroids and the Tutte polynomial
    • Higher dimensions
    • Appendices
    • Appendix A
    • Appendix B
  • Reviews
     
     
    • Each topic is described in a rigorous way such that advanced undergraduate mathematics students will enjoy them.

      Carlos Alejandro Alfaro, Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
2018; 325 pp
MSC: Primary 05;

Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces.

Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book.

Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, \(M\)-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.

Readership

Undergraduate and graduate students and researchers interested in games on graphs, Riemann surfaces, finite abelian groups, and Markov chains.

  • Divisors
  • The dollar game
  • The Laplacian
  • Algorithms for winning
  • Acyclic orientations
  • Riemann-Roch
  • Sandpiles
  • The sandpile group
  • Burning and duality
  • Threshold density
  • Topics
  • Trees
  • Harmonic morphisms
  • Divisors on complete graphs
  • More about sandpiles
  • Cycles and cuts
  • Matroids and the Tutte polynomial
  • Higher dimensions
  • Appendices
  • Appendix A
  • Appendix B
  • Each topic is described in a rigorous way such that advanced undergraduate mathematics students will enjoy them.

    Carlos Alejandro Alfaro, Mathematical Reviews
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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