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Algebraic Matroids in Applications by Zvi H Rosen a Dissertation Algebraic Matroids in Applications by Zvi H Rosen A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bernd Sturmfels, Chair Professor Lior Pachter Professor Yun S. Song Spring 2015 Algebraic Matroids in Applications Copyright 2015 by Zvi H Rosen 1 Abstract Algebraic Matroids in Applications by Zvi H Rosen Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Algebraic matroids are combinatorial objects defined by the set of coordinates of an algebraic variety. These objects are of interest whenever coordinates hold significance: for instance, when the variety describes solution sets for a real world problem, or is defined using some combinatorial rule. In this thesis, we discuss algebraic matroids, and explore tools for their computation. We then delve into two applications that involve algebraic matroids: probability matrices and tensors from statistics, and chemical reaction networks from biology. i In memory of my grandparents יצחק אהרN וחנה רייזל דייוויס ז"ל! Isaac and Ann Davis יצחק יוסP ושרה רוזN ז"ל! Isaac and Sala Rosen whose courage and perseverance through adversity will inspire their families for generations. ii Contents Contents ii List of Figures iv List of Tables v 1 Introduction 1 1.1 Summary of Main Results . 1 1.2 Examples . 2 1.3 Definitions, Axioms, and Notation . 5 2 Computation 10 2.1 Symbolic Algorithm . 11 2.2 Linear Algebra . 11 2.3 Sample Computations for Applications . 15 3 Statistics: Joint Probability Matroid 25 3.1 Completability of Partial Probability Matrix . 28 3.2 Semialgebraic Description . 35 3.3 Completion Algorithms . 39 3.4 Algebraic Matroids . 45 3.5 Generalizations . 47 4 Biology: Chemical Reaction Matroid 50 4.1 From Biology to Algebra . 53 4.2 Ideals, Varieties, and Nine Points . 57 4.3 Multistationarity and its Discriminant . 59 4.4 Algebraic Matroids and Parametrizations . 62 4.5 Polyhedral Geometry . 66 4.6 Parameter Estimation . 68 4.7 From Algebra to Biology . 73 5 Further Directions 76 iii Bibliography 79 iv List of Figures 1.1 Example of linear matroid. 3 1.2 Example of graphic matroid. 4 2.1 Schlegel diagram for the affine representation of M(PL4) . 16 2.2 Circuit degree frequency for M(PL4). ....................... 18 2.3 Base degree frequency for M(PL4).......................... 18 2.4 Non-cobases of M(Imix)................................ 19 2.5 Base degrees for M(Imix)............................... 19 2.6 Affine representation of the MAP kinase matroid. 21 2.7 Base of M(Gr(3; 6)) with base degree 7....................... 22 2.8 Circuit degree frequency for M(Gr(3; 6)). ..................... 23 2.9 Circuit of M(Gr(3; 6)) with degree 12........................ 23 2.10 Non-Pappus matroid. 23 3.1 Completable probability masks of 2 × 2 matrices with diagonal entries observed. 29 3.2 Completable probability masks of 3 × 3 matrices with diagonal entries observed. 31 3.3 Region defined by (3.11) inside the cube [0; 2]3. .................. 36 3.4 Solution curves for c = 1=9; 1=10; 1=16; 1=36; 1=64, and 1=150. 43 4.1 Graphic rep. of Mone, and affine rep. of the rank 4 component of Mpar. 70 4.2 Schematic diagram for systems biology research. 73 v List of Tables 2.1 Commands to compute algebraic matroids using matroids.m2 .......... 12 3.2 Algorithm for completing probability matrix projections. 41 4.1 The 19 species in the Wnt shuttle model. 55 4.2 The 31 reactions in the Wnt shuttle model. 56 4.3 Frequencies for the sampling schemes. 61 4.4 Frequencies for testing robustness scheme. 61 4.5 The 951 circuit polynomials, by numbers of unknowns xi and kj. 64 4.6 Reducing the steady state equations to the 2092 bases of base degree 1 . 65 vi Acknowledgments אוֹדה ה' בכל לבי אספרה כל נפלאותיK; (תהליM ט;ב)! I thank the Lord with my whole heart . (Psalms 9:2) Working towards a doctorate can come with unexpected challenges. The process involves learning a great deal about one’s area of research, as well as learning a lot about yourself. I have been blessed to have many supporters who encouraged me to pursue a career in math- ematics. Amitai Bin-Nun encouraged me to pursue research when I was an undergraduate, helping me apply to summer programs; he continues providing insightful advice when I call. Jessica Sidman gave me my first experience in math research, getting me hooked on the ex- perience; she also continues to be a great advisor. Many professors from my undergraduate training, especially Dennis DeTurck and Andreea Nicoara, taught me beautiful mathematics and helped me apply to graduate school. My mathematical life in Berkeley included terrific teaching experiences with Per Persson and Slobodan Simic. My graduate research experiences were fun and enriching thanks to all of my coauthors, especially Elizabeth Gross, Heather Harrington, and Kaie Kubjas, with whom I wrote the work forming much of this thesis. I also want to extend my gratitude to Carina Curto, Alex Fink, Heather Harrington, and Shmuel Onn for their help in the process of applying to postdoctoral grants and jobs. I owe a great debt to Bernd Sturmfels for taking me on as a student. It has been a privilege to benefit from his mathematical insight and to participate in the community of great thinkers that surrounds him. I also want to thank him for going through multiple drafts of all my writing including this dissertation, and providing detailed feedback to improve it. He has demonstrated tremendous patience and support, and I am incredibly grateful. The professors who served on the committee for my qualifying examination and my disser- tation were Mark Haiman, Lior Pachter, Satish Rao, Yun Song, and Dan-Virgil Voiculescu. Their willingness to participate is highly appreciated. The Berkeley math department has an incredible staff who make the lives of the graduate student population much easier. In par- ticular, I want to extend my appreciation to Barb Waller, Marsha Snow, and Judie Filomeo. I also spent a semester at the Max Planck Institute for Mathematics in Bonn, Germany, and Anke Völzmann was incredibly helpful in locating resources. Moving to Berkeley, CA from my home on the East Coast was a daunting enterprise in itself and I thank the members of Congregation Beth Israel, particularly Rabbi Yonatan and Frayda Cohen and Avraham and Ruchama Burrell whose hospitality and kindness to me has been nothing short of extraordinary. I also want to thank Maharat Victoria Sutton and Adam Brelow, Zeev and Tamara Neumeier, Naaman and Meechal Kam, and Maayan and Elishav Rabinovich for being so welcoming and generous. My group of friends, particularly Ali Austerlitz, Arabella Bangura, Benjamin Epstein, Boaz Haberman, Amalya Lehmann, Matty Lichtenstein, Shivaram Lingamneni and Jonathan Thirman made Berkeley feel like home, and gave me tremendous support and friendship. I also want to thank Eitan Adler, Yonatan Cantor, Jonathan Eskreis-Winkler, Elie Friedman, Miki Friedmann, Yoni Halpern, vii Ariella Meisel, Noam Pratzer, David Pruwer, and Mordechai Treiger whose support from afar kept me going. I could never have made it this far without the love and support of my family: my mother and father especially, for always making themselves available in times of need, and for serving as living examples for me. My siblings and my brothers-in-law have also been unfailingly kind and welcoming whenever I have needed their help. It is a wonderful gift to have family that always picks up the phone. 1 Chapter 1 Introduction Both chapters 1 and 2 are based on material from Computing Algebraic Matroids [48]. The definitions and introductory examples are brought to aid exposition, but similar examples can be found in standard texts such as [45] and [56]. Algebraic matroids have a surprisingly long history. They were studied as early as the 1930’s by van der Waerden, in his textbook Moderne Algebra [55, Chapter VIII], and MacLane in one of his earliest papers on matroids [35]. The topic lay dormant until the 70’s and 80’s, when a number of papers about representability of matroids as algebraic ma- troids were published: notably by Ingleton and Main [23], Dress and Lovasz [11], the thesis of Piff [46], and extensively by Lindström ([32, 33, 30, 31], among others). In recent years, the algebraic matroids of toric varieties found application (e.g. in [43]); however, they have been primarily confined to that setting. Renewed interest in algebraic matroids comes from the field of matrix completion, start- ing with [26], where the set of entries of a low-rank matrix are the ground set of the algebraic matroid associated to the determinantal variety. In applied algebra in general, coordinates typically carry real-world significance, and the matroid has inherent interest as the depen- dence structure among those quantities. Even for varieties arising in pure mathematics, distinguished coordinates may have combinatorial meaning, in which case the matroid also provides insight. 1.1 Summary of Main Results The purpose of this dissertation is to provide the relevant tools for computation of algebraic matroids and to actually use them in practice. The remainder of Chapter 1 introduces matroids to the reader and defines technical language that will be used in the thesis. We prove some basic results where the proof is particularly instructive or relevant. In Chapter 2, we summarize two techniques for computing algebraic matroids associated to prime ideals. An original definition in this chapter is the non-matroidal locus of an affine variety. This is a subvariety which exhibits degenerate behavior under coordinate projection.
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