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Softcover ISBN:  9781470442187 
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Softcover ISBN:  9781470442187 
Product Code:  MBK/114 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470448615 
Product Code:  MBK/114.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $60.00 
Softcover ISBN:  9781470442187 
eBook ISBN:  9781470448615 
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 Details2018; 325 ppMSC: Primary 05;
Divisors and Sandpiles provides an introduction to the combinatorial theory of chipfiring 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 graphtheoretic RiemannRoch 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 fixedenergy 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 chipfiring to other areas of mathematics, including the matrixtree 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.
ReadershipUndergraduate 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

RiemannRoch

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


Additional Material

Reviews

Each topic is described in a rigorous way such that advanced undergraduate mathematics students will enjoy them.
Carlos Alejandro Alfaro, Mathematical Reviews


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Divisors and Sandpiles provides an introduction to the combinatorial theory of chipfiring 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 graphtheoretic RiemannRoch 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 fixedenergy 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 chipfiring to other areas of mathematics, including the matrixtree 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.
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

RiemannRoch

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