COMBINATORIAL DIVISOR THEORY FOR GRAPHS A Thesis Presented to The Academic Faculty by Spencer Backman In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Algorithms, Combinatorics, and Optimization Georgia Institute of Technology School of Mathematics May 2014 Copyright c 2014 by Spencer Backman COMBINATORIAL DIVISOR THEORY FOR GRAPHS Approved by: Professor Matthew Baker, Professor Sebastian Pokutta Committee Chair, Advisor School of Industrial and Systems School of Mathematics Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Sergey Norin Professor Josephine Yu Reader School of Mathematics Department of Mathematics and Georgia Institute of Technology Statistics McGill University Professor Robin Thomas Date Approved: April 7th 2014 School of Mathematics Georgia Institute of Technology To my parents, Peter Backman and Annie Christopher, for their unwavering support. iii ACKNOWLEDGEMENTS First and foremost, I would like to express my deepest gratitude to my advisor Matthew Baker for his support and guidance during my graduate career. In ad- dition to the amazing mathematics which I have learned from Matt, I have received invaluable lessons about mathematical inquiry, in particular, the impossibly difficult art of asking the \right" questions. Many thanks to my fellow ACO and Mathematics students for their tangible and intangible assistance moving through my graduate career at Georgia Tech. In particular, I'd like to give a shout-out to Farbod Shokrieh, Luo Ye, Pedro Rangel, Sarah Miracle, Pushkar Tripathi, Arash Asadi, Noah Streib, Amanda Pascoe, Luke Postle, Daniel Dadush, Prateek Bhakta, Ying Xiao, Amey Kaloti, Robert Krone, Peter Whalen, and Chun-Hung Liu. I'd also like to express my profound appreciation for the Georgia Tech faculty from whom I learned what it is to be a mathematician, including but certainly not limited to Robin Thomas, Prasad Tetali, Ernie Croot, Asaf Shapira, Dana Randall, Anton Leykin, Josephine Yu, Greg Bleckherman, Brett Wick, and Sebastian Pokutta. Many thanks to the Mathematics Department at the University of California Berkeley for making me feel welcome during my time there, and for fueling my math- ematical growth. In particular, I'd like to express my sincere thanks to Bernd Sturm- fels, Lauren Williams, Jose Rodriguez, Mariah Monks, Zvi Rosen, Luke Oeding, Noah Giansiracusa, Elina Robeva, Florian Block, Felipe Rincon, Steven Sam, Chris Hillar, Atoshi Chowdhury, Harold Williams, Gus Schrader, Will Johnson, Sarah Brodsky, Alfredo Ch´avez, Thomas Kahle, Svante Linusson, Nathan Ilten, and Andrew Dudzik. Additional thanks go out to Madhusudan Manjunath, Sergey Norin, Lionel Levine, iv Gregg Musiker, David Perkinson, Diane Maclagan, Omid Amini, Federico Ardila, Sal Barone, Fatameh Mohammadi, Abraham Mart´ındel Campo, and Ethan Cotteril. My apologies to anyone I may have forgotten to include in this implicitly partial list. I have been unreasonably fortunate to have so much support from so many wonderful people, and I feel that I am bound to unintentionally omit someone's name. v TABLE OF CONTENTS DEDICATION .................................. iii ACKNOWLEDGEMENTS .......................... iv LIST OF FIGURES .............................. viii SUMMARY .................................... ix I INTRODUCTION ............................. 1 II RIEMANN-ROCH THEORY FOR GRAPH ORIENTATIONS . 10 2.1 Introduction . 10 2.2 Notation and Terminology . 15 2.3 Generalized Cycle, Cocycle, and Cycle-Cocyle Reversal Systems . 18 2.4 Oriented Dhar's Algorithm . 21 2.5 Directed Path Reversals and the Riemann-Roch Formula . 28 2.6 Luo's Theorem on Rank-Determining Sets . 34 2.7 Max-Flow Min-Cut and Divisor Theory . 38 2.8 Partial Orientations of Metric Graphs . 48 III TRANSFINITE CHIP-FIRING ..................... 52 3.1 Introduction . 52 3.2 Metric Chip-Firing and Reduced Divisors . 53 3.3 Infinite Greedy Reduction . 56 3.4 Running Time Analysis via Ordinal Numbers . 60 IV CHIP-FIRING VIA OPEN COVERS ................. 66 4.1 Introduction . 66 4.2 Notation and Terminology . 68 4.3 Preliminary Results for Discrete Graphs . 70 4.4 Spanning Tree Bijection . 75 4.5 Chip-firing via Open Covers of Metric Graphs . 81 vi V RIEMANN-ROCH THEORY FOR DIRECTED GRAPHS AND ARITHMETICAL GRAPHS ...................... 84 5.1 Introduction . 84 5.1.1 Basic Notations and Definitions . 87 5.2 Riemann-Roch Theory for Sub-lattices of ΛR ............. 88 5.2.1 Main Theorems . 88 5.2.2 Amini and Manjunath's Riemann-Roch theory for lattices . 91 5.2.3 Wilmes' Lattice Reduction Algorithm . 103 5.3 Chip-Firing Games on Directed Graphs . 105 5.3.1 Row Chip-Firing Game, The Sandpile Model, and Riemann- Roch Theory . 105 5.3.2 Column Chip-Firing Game, G~ -Parking Functions, and Riemann- Roch Theory . 116 5.4 Arithmetical Graphs . 119 5.4.1 A Combinatorial Proof of Lorenzini's Theorem . 119 5.4.2 Arithmetical Graphs with the Riemann-Roch Property . 126 5.4.3 Arithmetical Graphs without the Riemann-Roch Property . 134 REFERENCES .................................. 139 VITA ........................................ 145 vii LIST OF FIGURES 1 A partial orientation with (a) an edge pivot, (b) a cocycle reversal, and (c) a cycle reversal. 14 2 A Jacob's ladder cascade. 19 3 The unfurling algorithm applied to the partial orientation on the top left, terminating with the acyclic partial orientation on the bottom right. 24 4 A sequence of equivalent partial orientations. The left and right par- tial orientations are both q-connected, but have different associated divisors. The partial orientation on the right is a directed spanning unicycle. 32 5 A directed path whose reversal produces an acyclic orientation. By Theorem 2.5.7 it follows that the divisor associated to the top orienta- tion has rank 0. 33 6 A full orientation of a metric graph and two other orientations obtained by \pushing" the change of orientation along the middle edge to the right and left. The push to the right causes directed cycles to appear while the push to the left does not. 35 7 Top: A network with source s, sink t, capacities listed next to edges, and a minimum cut of size 4 colored red. Bottom: A maximum flow on this network with flow value 4. Note that the flow along each edge in the minimum cut is equal to the capacity of that edge. 39 8 (a) Left: An orientation of a graph G Right: The divisor DO on G, (b) A divisor D on G, (c)O The network N, with auxiliary vertices s and t. The set S is colored blue, the set T is colored green, and the additional edges are labeled by their capacities. The remaining edges have capacity 1. The edges colored red are the support of a maximal flow f. (d) An orientation D obtained by reversing the flow f on N and then restricting to G.........................O 42 9 A picture proof that for partially orientable divisors,χ ¯(S1;D) =χ ¯(S2;D) = 0 impliesχ ¯(S1 S2;D) =χ ¯(S1 S2;D) = 0: This figure does not im- mediately apply[ in the proof of\ Theorem 2.7 because we cannot pre- suppose the divisors in question are partially orientable, although it c can be converted into a proof if we contract G[(S1 S2) ] and reduce to the case of full orientations. .[ . 47 viii SUMMARY Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the develop- ment of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cor- nerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as \combinatorial shadows" of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connec- tions to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry. ix CHAPTER I INTRODUCTION Chip-firing is a simple and elegant graph theoretic process with connections to var- ious areas of mathematics, and the sciences at-large. For describing chip-firing, we begin with a finite set of chips on the vertices of a graph. The fundamental operation is firing, whereby a vertex sends a chip to each of its neighbors and loses its degree number of chips in the process, so that the total number of chips in the graph is conserved. We remark that our use of the word chips is intended to connote a collec- tion indistinguishable poker chips sitting at the vertices of the graph, as opposed to computer chips (although rather interestingly, the latter interpretation is not without merit, e.g. [17]). If one encodes a chip configuration by a vector ~x then the operation of firing the ith vertex can described in a linear algbraic fashion by subtracting the ith column of the Laplacian matrix from ~x.