PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS A Dissertation presented to the Faculty of the Graduate School, University of Missouri, Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by KILLIAN MEEHAN Dr. Calin Chindris, Dissertation Supervisor Dr. Jan Segert, Dissertation Supervisor MAY 2018 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS presented by Killian Meehan, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Associate Professor Calin Chindris Associate Professor Jan Segert Assistant Professor David C. Meyer Associate Professor Mihail Popescu ACKNOWLEDGEMENTS To my thesis advisors, Calin Chindris and Jan Segert, for your guidance, humor, and candid conversations. David Meyer, for our emphatic sharing of ideas, as well as the savage question- ing of even the most minute assumptions. Working together has been an absolute blast. Jacob Clark, Brett Collins, Melissa Emory, and Andrei Pavlichenko for conver- sations both professional and ridiculous. My time here was all the more fun with your friendships and our collective absurdity. Adam Koszela and Stephen Herman for proving that the best balm for a tired mind is to spend hours discussing science, fiction, and intersection of the two. My brothers, for always reminding me that imagination and creativity are the loci of a fulfilling life. My parents, for teaching me that the best ideas are never found in an intellec- tual vacuum. My grandparents, for asking me the questions that led me to where I am. ii Contents Acknowledgements ii Abstract vi 1 Introduction1 2 Preliminaries3 2.1 Category Theory..............................3 2.1.1 Universal Properties: Kernels and Cokernels..........5 2.1.2 Additive and Krull-Schmidt Categories.............9 2.2 Quiver Theory................................ 11 2.2.1 The Path Algebra.......................... 13 2.2.2 Quivers with Relations....................... 14 2.2.3 Representation Type........................ 15 2.2.4 Equivalence of Categories..................... 17 2.2.5 Auslander-Reiten Theory..................... 17 2.3 Persistent Homology............................ 18 2.3.1 Simplicial Complexes....................... 18 2.3.2 Simplicial Homology........................ 19 2.3.3 GPMs................................. 22 iii 2.3.4 Process................................ 23 2.4 Stability.................................... 25 2.5 Interleaving Metric............................. 25 2.5.1 Fixed Points............................. 27 2.5.2 Submodules and Quotient Modules............... 28 2.5.3 Weights and Suspension at Infinity................ 31 2.6 Bottleneck Metric.............................. 33 2.7 The Category Generated by Convex Modules.............. 35 3 Categorical Framework 37 3.1 Motivation.................................. 37 3.2 Manifestations of the Structural Theorem................ 37 3.2.1 Introduction............................. 37 3.2.2 Matrix Structural Theorem.................... 45 3.2.3 Categorical Structural Theorem and Structural Equivalence. 50 3.3 Proving the Categorical Structural Theorem............... 51 3.3.1 Persistence Objects and Filtered Objects............. 51 3.3.2 Chain Complexes and Filtered Chain Complexes....... 54 3.4 Categorical Frameworks for Persistent Homology........... 57 3.4.1 Standard Framework using Persistence Vector Spaces..... 57 3.4.2 Alternate Framework using Quotient Categories........ 59 3.5 Proving Structural Equivalence...................... 62 3.5.1 Forward Structural Equivalence................. 62 3.5.2 Reverse Structural Equivalence.................. 72 iv 3.6 Bruhat Uniqueness Lemma........................ 75 3.7 Constructively Proving the Matrix Structural Theorem........ 77 3.7.1 Linear Algebra of Reduction................... 77 3.7.2 Matrix Structural Theorem via Reduction............ 81 4 An Isometry Theorem for Generalized Persistence Modules 87 4.1 Motivation.................................. 87 4.1.1 Algebraic Stability......................... 87 4.1.2 Connection to Finite-dimensional Algebras........... 88 4.2 A Particular Class of Posets........................ 90 4.3 Homomorphisms and Translations.................... 99 4.4 Isometry Theorem for Finite Totally Ordered Sets........... 110 4.5 Proof of Main Results............................ 120 4.6 Examples................................... 127 5 The Interleaving Distance as a Limit 137 5.1 Motivation.................................. 137 5.2 Restriction and Inflation.......................... 138 5.3 The Shift Isometry Theorem........................ 143 5.4 Interleaving Distance as a Limit...................... 150 5.5 Regularity.................................. 156 Bibliography 161 Vita 167 v ABSTRACT The purpose of this thesis is to advance the study and application of the field of persistent homology through both categorical and quiver theoretic viewpoints. While persistent homology has its roots in these topics, there is a wealth of material that can still be offered up by using these familiar lenses at new angles. There are three chapters of results. Chapter3 discusses a categorical framework for persistent homology that cir- cumvents quiver theoretic structure, both in practice and in theory, by means of viewing the process as factored through a quotient category. In this chapter, the widely used persistent homology algorithm collectively known as reduction is pre- sented in terms of a matrix factorization result. The remaining results rest on a quiver theoretic approach. Chapter4 focuses on an algebraic stability theorem for generalized persistence modules for a certain class of finite posets. Both the class of posets and their dis- cretized nature are what make the results unique, while the structure is taken with inspiration from the work of Ulrich Bauer and Michael Lesnick. Chapter5 deals with taking directed limits of posets and the subsequent expan- sion of categories to show that the discretized work in the second section recovers classical results over the continuum. vi Chapter 1 Introduction This research is motivated by the crossover between pure mathematics and real world problem solving. I have focused largely on topological methods of analyz- ing data, and in particular the device of persistent homology [ZC05a], [EH10]. At its inception, this field gathered algebraic topology, category theory, and quiver the- ory to provide a new theory of robust analysis of data that is independent of scale. Even in the short time since its inception, persistent homology has grown to give and take from a staggering number of disciplines—mathematical and otherwise. All the results featured in this manuscript are the result of collaborative efforts. The nature of the material and the coauthors with which it was derived is split into two groups. The first result (Chapter3) is from the paper [MPS17] and was written with my dissertation advisor Jan Segert and fellow graduate student Andrei Pavlichenko. We show that the category of filtered chain complexes is in fact Krull-Schmidt, and that this allows for decomposition before applying the homology functor. Fur- thermore, this abstract result—which we name the Categorical Structural Theo- rem—constitutes a non-constructive proof of the Matrix Structural Theorem, which is the matrix factorization result that is key to reduction, a common persistent ho- 1 mology algorithm in the literature. The Categorical Structural Theorem is the foundation for an alternate workflow for the persistent homology process that cir- cumvents quiver theoretic decomposition results entirely, using instead a quotient category to the category of filtered chain complexes. The last three chapters of results were all obtained in collaboration with post- doctoral researcher David Meyer. They stem from the papers [MM17a] (see Chap- ter4) and [MM17b] (see Chapter5), and Chapter 6 is the result of our ongoing re- search. All these results are related to algebraic stability between the interleaving metric on persistence modules and bottleneck metrics on barcodes. One overarch- ing goal of all three chapters is to bypass the wall raised by quiver theory on the road to multi-dimensional persistence: that the immaculate decomposition result over the An quiver fails for all but a very small list of additional quivers. In par- ticular, the posets that would be used for multi-dimensional persistence have, as quivers, representation theory that is known to be unsolvable. Through discretiza- tion of the usual R-indexed persistent homology and a restriction on the permitted category of indecomposables, we obtain algebraic stability results for a large class of finite posets that are not totally ordered. 2 Chapter 2 Preliminaries 2.1 Category Theory Definition 2.1.1. A category is: C a class of objects, Obj( ), which we will simply denote as itself, • C C a set of morphisms Hom(X; Y ) for all pairs of objects X; Y , • 2 C an identity morphism 1x Hom(X; X) for all X , and • 2 2 C a composition map • Hom(X; Y ) Hom(Y; Z) Hom(X; Z) × ! for all triplets X; Y; Z , such that 2 C 1. for all φ Hom(X; Y ); 1y φ = φ = φ 1x, and 2 ◦ ◦ 2. for all f g W X Y h Z; −! −! −! h (g f) = (h g) f. ◦ ◦ ◦ ◦ Definition 2.1.2. We define the following morphism types: 3 A category with zero morphisms is one