2018;
325 pp;
Softcover
MSC: Primary 05;
Print ISBN: 978-1-4704-4218-7
Product Code: MBK/114
List Price: $79.00
AMS Member Price: $63.20
MAA Member Price: $71.10
Electronic ISBN: 978-1-4704-4861-5
Product Code: MBK/114.E
List Price: $79.00
AMS Member Price: $63.20
MAA Member Price: $71.10
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Supplemental Materials
Divisors and Sandpiles: An Introduction to Chip-Firing
Share this pageScott Corry; David Perkinson
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
Table of Contents
Divisors and Sandpiles: An Introduction to Chip-Firing
- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Part 1 . Divisors 116
- Part 2 . Sandpiles 85100
- Part 3 . Topics 159174
- Appendices 295310
- Back Cover Back Cover1342