{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/117780"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/117780","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Root system chip-firing","abstract":"This thesis investigates an extension of the classical chip-firing process to \"other Cartan-Killing types.\" In Chapter 1 we review the classical chip-firing game: the states of this process are configurations of chips on the vertices of a graph; the transition moves are firings whereby a vertex with at least as many chips as neighbors may send one chip to each neighbor. A fundamental property of chip-firing is that it is confluent: from any initial configuration, all sequences of firings lead to the same terminal configuration. In Chapter 2 we discuss Propp's labeled chip-firing process on the infinite path, for which confluence becomes a subtler question. We prove that labeled chip-firing is confluent starting from an even number of chips at the origin (but not from an odd number). In Chapter 3 we reinterpret labeled chip-firing as a process on the weight lattice of a root system, where the firing moves consist of adding a positive root whenever the weight we are at is orthogonal to that root. We call this the central-firing process. We give conjectures about certain initial weights from which central-firing is confluent. We also prove that central-firing is always confluent from all initial weights if we mod out by the action of the Weyl group, thereby giving a generalization of unlabeled chip firing on the infinite path to other types. In Chapter 4 we introduce some remarkable deformations of the central-firing process which we call the symmetric and truncated interval-firing processes. These are analogous to the Catalan and Shi hyperplane arrangements. We prove that these interval-firing processes are always confluent from all initial weights. In Chapter 5 we study the set of weights with given interval-firing stabilization. We show that the number of weights with given stabilization is a polynomial in our deformation parameter. We call these polynomials the symmetric and truncated Ehrhart-like polynomials, because they are analogous to the Ehrhart polynomial of a polytope. We conjecture that the Ehrhart-like polynomials have nonnegative integer coefficients. In Chapter 6 we prove \"half\" of this positivity conjecture by providing an explicit, positive formula for the symmetric Ehrhart-like polynomials.","abstract_html":"This thesis investigates an extension of the classical chip-firing process to "other Cartan-Killing types." In Chapter 1 we review the classical chip-firing game: the states of this process are configurations of chips on the vertices of a graph; the transition moves are firings whereby a vertex with at least as many chips as neighbors may send one chip to each neighbor. A fundamental property of chip-firing is that it is confluent: from any initial configuration, all sequences of firings lead to the same terminal configuration. In Chapter 2 we discuss Propp's labeled chip-firing process on the infinite path, for which confluence becomes a subtler question. We prove that labeled chip-firing is confluent starting from an even number of chips at the origin (but not from an odd number). In Chapter 3 we reinterpret labeled chip-firing as a process on the weight lattice of a root system, where the firing moves consist of adding a positive root whenever the weight we are at is orthogonal to that root. We call this the central-firing process. We give conjectures about certain initial weights from which central-firing is confluent. We also prove that central-firing is always confluent from all initial weights if we mod out by the action of the Weyl group, thereby giving a generalization of unlabeled chip firing on the infinite path to other types. In Chapter 4 we introduce some remarkable deformations of the central-firing process which we call the symmetric and truncated interval-firing processes. These are analogous to the Catalan and Shi hyperplane arrangements. We prove that these interval-firing processes are always confluent from all initial weights. In Chapter 5 we study the set of weights with given interval-firing stabilization. We show that the number of weights with given stabilization is a polynomial in our deformation parameter. We call these polynomials the symmetric and truncated Ehrhart-like polynomials, because they are analogous to the Ehrhart polynomial of a polytope. We conjecture that the Ehrhart-like polynomials have nonnegative integer coefficients. In Chapter 6 we prove "half" of this positivity conjecture by providing an explicit, positive formula for the symmetric Ehrhart-like polynomials.","abstract_has_math":false,"creators":["Hopkins, Samuel F"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics.","school":null,"contributors":[],"advisors":["Alexander Postnikov."],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018","date_published":"2018","updated_at":"2026-07-22T22:22:13Z","subjects":["Mathematics."],"languages":["eng"],"rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/117780","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Alexander Postnikov."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Department of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Hopkins, Samuel F"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-09-17T14:49:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-09-17T14:49:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2018"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/117780"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.","Cataloged from student-submitted PDF version of thesis.","Includes bibliographical references (pages 195-200)."]},{"key":"dc:description.abstract","label":"Abstract","values":["This thesis investigates an extension of the classical chip-firing process to \"other Cartan-Killing types.\" In Chapter 1 we review the classical chip-firing game: the states of this process are configurations of chips on the vertices of a graph; the transition moves are firings whereby a vertex with at least as many chips as neighbors may send one chip to each neighbor. A fundamental property of chip-firing is that it is confluent: from any initial configuration, all sequences of firings lead to the same terminal configuration. In Chapter 2 we discuss Propp's labeled chip-firing process on the infinite path, for which confluence becomes a subtler question. We prove that labeled chip-firing is confluent starting from an even number of chips at the origin (but not from an odd number). In Chapter 3 we reinterpret labeled chip-firing as a process on the weight lattice of a root system, where the firing moves consist of adding a positive root whenever the weight we are at is orthogonal to that root. We call this the central-firing process. We give conjectures about certain initial weights from which central-firing is confluent. We also prove that central-firing is always confluent from all initial weights if we mod out by the action of the Weyl group, thereby giving a generalization of unlabeled chip firing on the infinite path to other types. In Chapter 4 we introduce some remarkable deformations of the central-firing process which we call the symmetric and truncated interval-firing processes. These are analogous to the Catalan and Shi hyperplane arrangements. We prove that these interval-firing processes are always confluent from all initial weights. In Chapter 5 we study the set of weights with given interval-firing stabilization. We show that the number of weights with given stabilization is a polynomial in our deformation parameter. We call these polynomials the symmetric and truncated Ehrhart-like polynomials, because they are analogous to the Ehrhart polynomial of a polytope. We conjecture that the Ehrhart-like polynomials have nonnegative integer coefficients. In Chapter 6 we prove \"half\" of this positivity conjecture by providing an explicit, positive formula for the symmetric Ehrhart-like polynomials."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Root system chip-firing"]}]}],"canonical_facts":{"dc:contributor.advisor":["Alexander Postnikov."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:creator":["Hopkins, Samuel F"],"dc:date.accessioned":["2018-09-17T14:49:19Z"],"dc:date.available":["2018-09-17T14:49:19Z"],"dc:date.issued":["2018"],"dc:description":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.","Cataloged from student-submitted PDF version of thesis.","Includes bibliographical references (pages 195-200)."],"dc:description.abstract":["This thesis investigates an extension of the classical chip-firing process to \"other Cartan-Killing types.\" In Chapter 1 we review the classical chip-firing game: the states of this process are configurations of chips on the vertices of a graph; the transition moves are firings whereby a vertex with at least as many chips as neighbors may send one chip to each neighbor. A fundamental property of chip-firing is that it is confluent: from any initial configuration, all sequences of firings lead to the same terminal configuration. In Chapter 2 we discuss Propp's labeled chip-firing process on the infinite path, for which confluence becomes a subtler question. We prove that labeled chip-firing is confluent starting from an even number of chips at the origin (but not from an odd number). In Chapter 3 we reinterpret labeled chip-firing as a process on the weight lattice of a root system, where the firing moves consist of adding a positive root whenever the weight we are at is orthogonal to that root. We call this the central-firing process. We give conjectures about certain initial weights from which central-firing is confluent. We also prove that central-firing is always confluent from all initial weights if we mod out by the action of the Weyl group, thereby giving a generalization of unlabeled chip firing on the infinite path to other types. In Chapter 4 we introduce some remarkable deformations of the central-firing process which we call the symmetric and truncated interval-firing processes. These are analogous to the Catalan and Shi hyperplane arrangements. We prove that these interval-firing processes are always confluent from all initial weights. In Chapter 5 we study the set of weights with given interval-firing stabilization. We show that the number of weights with given stabilization is a polynomial in our deformation parameter. We call these polynomials the symmetric and truncated Ehrhart-like polynomials, because they are analogous to the Ehrhart polynomial of a polytope. We conjecture that the Ehrhart-like polynomials have nonnegative integer coefficients. In Chapter 6 we prove \"half\" of this positivity conjecture by providing an explicit, positive formula for the symmetric Ehrhart-like polynomials."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/117780"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Root system chip-firing"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:13Z"}