Overview

Full Product Details

Author:   Caroline J. Klivans
Publisher:   Taylor & Francis Ltd
Imprint:   CRC Press
Weight:   0.421kg
ISBN:  

9781138634091

ISBN 10:   1138634093
Pages:   296
Publication Date:   21 November 2018
Audience:   Professional and scholarly ,  College/higher education ,  Professional & Vocational ,  Tertiary & Higher Education
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

Introduction A brief introduction. Origins/History. Chip-firing on Finite Graphs The chip-firing process. Confluence. Stabilization. Toppling time. Stabilization with a sink. Long-term stable configurations. The sandpile Markov chain. Spanning Trees Spanning trees. Statistics on Trees. Merino’s Theorem. Cori-Le Borgne bijection. Acyclic orientations. Parking functions. Dominoes. Avalanche polynomials. Sandpile Groups Toppling dynamics. Group of chip-firing equivalence. Identity. Combinatorial invariance. Sandpile groups and invariant factors. Discriminant groups. Sandpile torsors. Pattern Formation Compelling visualizations. Infinite graphs. The one-dimensional grid. Labeled chip-firing. Two and more dimensional grids. Other lattices. The identity element. Avalanche Finite Systems M-matrices. Chip-firing on M-matrices. Stability. Burning. Directed graphs. Cartan matrices as M-matrices. M-pairings. Higher Dimensions An illustrative example. Cell complexes. Combinatorial Laplacians. Chip-firing in higher dimensions. The sandpile group. Higher-dimensional trees. Sandpile groups. Cuts and flows. Stability. Divisors Divisors on curves. The Picard group and Abel-Jacobi theory. Riemann-Roch Theorems. Torelli’s Theorem. The Pic^g (G) torus. Metric graphs and tropical geometry. Arithmetic geometry. Arithmetical graphs. Riemann-Roch for lattices. Two variable zeta-functions. Enumerating arithmetical structures. Ideals Ideals. Toppling ideals. Tree ideals. Resolutions. Critical ideals. Riemann-Roch for monomial ideals.

Reviews

Author Information

Caroline J. Klivans received a BA degree in mathematics from Cornell University and a PhD in applied mathematics from MIT. Currently, she is an Associate Professor in the Division of Applied Mathematics at Brown University. She is also an Associate Director of ICERM (Institute for Computational and Experimental Research in Mathematics). Before coming to Brown she held positions at MSRI, Cornell and the University of Chicago. Her research is in algebraic, geometric and topological combinatorics.

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions