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University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2015 Polyhedral Problems in Combinatorial Convex Geometry Liam Solus University of Kentucky, [email protected] Digital Object Identifier: http://dx.doi.org/10.13023/ETD.2016.014 Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Solus, Liam, "Polyhedral Problems in Combinatorial Convex Geometry" (2015). Theses and Dissertations-- Mathematics. 32. https://uknowledge.uky.edu/math_etds/32 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Liam Solus, Student Dr. Benjamin Braun, Major Professor Dr. Peter Perry, Director of Graduate Studies POLYHEDRAL PROBLEMS IN COMBINATORIAL CONVEX GEOMETRY DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Liam Solus Lexington, Kentucky Co-Directors: Dr. Benjamin Braun, Professor of Mathematics, and Dr. Carl Lee, Professor of Mathematics Lexington, Kentucky 2015 Copyright c Liam Solus 2015 ABSTRACT OF DISSERTATION POLYHEDRAL PROBLEMS IN COMBINATORIAL CONVEX GEOMETRY In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem. In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a triangulation of each r-stable (n,k)-hypersimplex within. We then use this trian- gulation to compute the facet-defining inequalities of these polytopes. In the k=2 case, we use shelling techniques to devise a combinatorial interpretation of the coef- ficients of the h*-polynomials in terms of independent sets of certain graphs. From this, we then extract some results on unimodality. We also characterize the Goren- stein r-stable (n,k)-hypersimplices, and we conclude that these also have unimodal h*-polynomials. In the second case, for a graph G on p vertices we consider the closure of the cone of concentration matrices of G. The extreme rays of this cone, and their associated ranks, have applications in maximum likelihood estimation for the undirected Gaus- sian graphical model associated to G. Consequently, the extreme ranks of this cone have been well-studied. Yet, there are few graph classes for which all the possible extreme ranks are known. We show that the facet-normals of the cut polytope of G can serve to identify extreme rays of this nonpolyhedral cone. We see that for graphs without K5 minors each facet-normal of the cut polytope identifies an extreme ray in the cone, and we determine the rank of this extreme ray. When the graph is also series-parallel, we find that all possible extreme ranks arise in this fashion, thereby extending the collection of graph classes for which all the possible extreme ranks are known. KEYWORDS: r-stable hypersimplices, hypersimplices, cut polytope, graphical mod- els, facet-ray identification. Author's signature: Liam Solus Date: December 3, 2015 iii POLYHEDRAL PROBLEMS IN COMBINATORIAL CONVEX GEOMETRY By Liam Solus Co-Director of Dissertation: Benjamin Braun Co-Director of Dissertation: Carl Lee Director of Graduate Studies: Peter Perry Date: December 3, 2015 For Mom, Dad, and Dan. ACKNOWLEDGMENTS First I would like to thank my advisor Ben Braun for his constant support and guidance. Ben genuinely loves to learn mathematics, and he inspires this passion in every one of his students. His love for mathematics, his social spirit, and creative mind inspired me to set my creative side free on all of my research problems and to unabashedly share my discoveries and the excitement they bring me. Ben generated numerous opportunities for me to pursue my mathematical interests and connect with the research community. I am so grateful to Ben for sharing with me his passion for algebraic and geometric combinatorics, as well as the ability to identify and initiate fruitful collaborations with other mathematicians. His advice has not only unlocked doors for me to explore my passions in mathematics, it has created opportunities for me to explore the world at large. Further, I would like to thank Carl Lee and Uwe Nagel for their support as members of my committee. I enjoyed many interesting conversations with both Carl and Uwe that provided me with useful insights and intuitions. I would also like to express a deep gratitude to Rudy Yoshida. Rudy's happy and outgoing personality couples with her strong abilities as a mathematician and statistician to make her a really excellent collaborator. I am so grateful to Rudy for offering me the opportunity to work on a project that introduced me to a whole field of study. Her collaboration truly allowed me to broaden my mathematical horizons, and it generated exciting opportunities for me to meet new collaborators. I am also very grateful to Caroline Uhler for her collaboration and support. Car- oline is a fantastic researcher, and working with her inspired me to truly push my limits. I am very grateful to her both for her collaboration on exciting projects, as well as her mentorship on integrating into the mathematical community. She has not iii only engaged my mathematical interests, but she has shown me invaluable techniques for developing new collaborations and generating new opportunities for me to pursue my dreams. I am also thankful to Takayuki Hibi for his collaboration and advisement during my time at Osaka University. Takayuki has an excellent intuition for combinatorial geometry, and his insights resulted in a fantastic collaboration. He also provided me with ample opportunities to present my research, and this really allowed me to develop my skills as a speaker. I am also grateful to the National Science Foundation East Asia and Pacific Summer Schools for U.S. Graduate Students and the Japan Society for the Promotion of Science for funding our collaboration at Osaka University. Similarly, I would like to thank the University of Kentucky Department of Mathematics and Department of Statistics for providing summer support for my research and travels. Also, I would like to thank the faculty of the Oberlin College mathematics depart- ment for helping me to discover my passion for mathematics. In particular, I would like to thank Kevin Woods for his advisement and encouragement to work with Ben at the University of Kentucky. As well, I would like to thank my high school Calcu- lus teacher, Cindy Howes, for first encouraging me to ask \why?" in a mathematics classroom. Finally, I would like to thank my family and friends for their love and support as I pursued my PhD. Thank you to my mom, dad, and brother for always being their to talk with me and offer their words of support as I chased my dreams. My mom and dad always encouraged me to pursue my passions, and they worked hard to generate amazing opportunities for me to learn and grow. I owe all the happiness I gain from my research to my parents' endless nuturing of my creativity. As well, thanks to all of my friends for being there to work on mathematics and relax when we could find the time! iv TABLE OF CONTENTS Acknowledgments . iii Table of Contents . v List of Figures . vi Chapter 1 Introduction . 1 1.1 Polyhedra . 2 1.2 Ehrhart Theory for Lattice Polytopes . 8 1.3 Spectrahedra and Spectrahedral Shadows . 13 1.4 Gaussian Graphical Models . 20 Chapter 2 The r-stable Hypersimplices . 23 2.1 Hypersimplices and r-stable Hypersimplices . 23 2.2 A Regular Unimodular Triangulation . 23 2.3 The Facets of the r-stable Hypersimplices .
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