Convex Polytopes, Algebraic Geometry, and Combinatorics
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Combinatorial and Discrete Problems in Convex Geometry
COMBINATORIAL AND DISCRETE PROBLEMS IN CONVEX GEOMETRY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Matthew R. Alexander September, 2017 Dissertation written by Matthew R. Alexander B.S., Youngstown State University, 2011 M.A., Kent State University, 2013 Ph.D., Kent State University, 2017 Approved by Dr. Artem Zvavitch , Co-Chair, Doctoral Dissertation Committee Dr. Matthieu Fradelizi , Co-Chair Dr. Dmitry Ryabogin , Members, Doctoral Dissertation Committee Dr. Fedor Nazarov Dr. Feodor F. Dragan Dr. Jonathan Maletic Accepted by Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS LIST OF FIGURES . vi ACKNOWLEDGEMENTS ........................................ vii NOTATION ................................................. viii I Introduction 1 1 This Thesis . .2 2 Preliminaries . .4 2.1 Convex Geometry . .4 2.2 Basic functions and their properties . .5 2.3 Classical theorems . .6 2.4 Polarity . .8 2.5 Volume Product . .8 II The Discrete Slicing Problem 11 3 The Geometry of Numbers . 12 3.1 Introduction . 12 3.2 Lattices . 12 3.3 Minkowski's Theorems . 14 3.4 The Ehrhart Polynomial . 15 4 Geometric Tomography . 17 4.1 Introduction . 17 iii 4.2 Projections of sets . 18 4.3 Sections of sets . 19 4.4 The Isomorphic Busemann-Petty Problem . 20 5 The Discrete Slicing Problem . 22 5.1 Introduction . 22 5.2 Solution in Z2 ............................................. 23 5.3 Approach via Discrete F. John Theorem . 23 5.4 Solution using known inequalities . 26 5.5 Solution for Unconditional Bodies . 27 5.6 Discrete Brunn's Theorem . -
A Logical Approach to Asymptotic Combinatorics I. First Order Properties
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATI-EMATlCS 65, 65-96 (1987) A Logical Approach to Asymptotic Combinatorics I. First Order Properties KEVIN J. COMPTON ’ Wesleyan University, Middletown, Connecticut 06457 INTRODUCTION We shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics; this framework provides methods for determining the probability that a large, finite structure, ran- domly chosen from a given class, will have a given property. Our main concern is the asymptotic probability: the limiting value as the size of the structure increases. For example, a common problem in elementary probability texts is to show that the asymptotic probabity that a per- mutation will have no fixed point is l/e. We shall say nothing about the closely related problem of determining rates of convergence, although the methods presented here may extend to such problems. To develop a general approach we must fix a language for specifying properties of structures. Thus, our approach is logical; logic is the branch of mathematics that deals with problems of language. In this paper we con- sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties. In the sequel to this paper we consider properties expressible in the more general language of monadic second order logic. Clearly, we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant. Therefore, we choose to consider only classes closed under disjoint unions and components (see Sect. -
Research Statement
RESEARCH STATEMENT SERGII MYROSHNYCHENKO Contents 1. Introduction 1 2. Nakajima-S¨ussconjecture and its functional generalization 2 3. Visual recognition of convex bodies 4 4. Intrinsic volumes of ellipsoids and the moment problem 6 5. Questions of geometric inequalities 7 6. Analytic permutation testing 9 7. Current work and future plans 10 References 12 1. Introduction My research interests lie mainly in Convex and Discrete Geometry with applications of Probability Theory and Harmonic Analysis to these areas. Convex geometry is a branch of mathematics that works with compact convex sets of non-empty interior in Euclidean spaces called convex bodies. The simple notion of convexity provides a very rich structure for the bodies that can lead to surprisingly simple and elegant results (for instance, [Ba]). A lot of progress in this area has been made by many leading mathematicians, and the outgoing work keeps providing fruitful and extremely interesting new results, problems, and applications. Moreover, convex geometry often deals with tasks that are closely related to data analysis, theory of optimizations, and computer science (some of which are mentioned below, also see [V]). In particular, I have been interested in the questions of Geometric Tomography ([Ga]), which is the area of mathematics dealing with different ways of retrieval of information about geometric objects from data about their different types of projections, sections, or both. Many long-standing problems related to this topic are quite intuitive and easy to formulate, however, the answers in many cases remain unknown. Also, this field of study is of particular interest, since it has many curious applications that include X-ray procedures, computer vision, scanning tasks, 3D printing, Cryo-EM imaging processes etc. -
Interactions of Computational Complexity Theory and Mathematics
Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require. -
Middleware and Application Management Architecture
Middleware and Application Management Architecture vorgelegt vom Diplom-Informatiker Sven van der Meer aus Berlin von der Fakultät IV – Elektrotechnik und Informatik der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften - Dr.-Ing. - genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Gorlatch Berichter: Prof. Dr. Dr. h.c. Radu Popescu-Zeletin Berichter: Prof. Dr. Kurt Geihs Tag der wissenschaftlichen Aussprache: 25.09.2002 Berlin 2002 D 83 Middleware and Application Management Architecture Sven van der Meer Berlin 2002 Preface Abstract This thesis describes a new approach for the integrated management of distributed networks, services, and applications. The main objective of this approach is the realization of software, systems, and services that address composability, scalability, reliability, and robustness as well as autonomous self-adaptation. It focuses on middleware for management, control, and use of fully distributed resources. The term integra- tion refers to the task of unifying existing instrumentations for middleware and management, not their replacement. The rationale of this work stems from the fact, that the current situation of middleware and management systems can be described with the term interworking. The actually needed integration of management and middleware concepts is still an open issue. However, identified trends in communications and computing demand for integrated concepts rather than the definition of new interworking scenarios. Distributed applications need to be prepared to be used and operated in a stable, secure, and efficient way. Middleware and service platforms are employed to solve this task. To guarantee this objective for a long-time operation, the systems needs to be controlled, administered, and maintained in its entirety, supporting the general aim of the system, and for each individual component, to ensure that each part of the system functions perfectly. -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical -
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its appli- cations. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators. The intellectual merits of this project include the development of applications of al- gebraic geometry and of combinatorial and combinatorial aspects of algebraic geometry. Specifically, Sottile will work to develop the theory and properties of orbitopes from the perspective of convex algebraic geometry, continue to investigate linear precision in geo- metric modeling, and apply the quantum Schubert calculus to linear systems theory. -
Combinatorics and Graph Theory I (Math 688)
Combinatorics and Graph Theory I (Math 688). Problems and Solutions. May 17, 2006 PREFACE Most of the problems in this document are the problems suggested as home- work in a graduate course Combinatorics and Graph Theory I (Math 688) taught by me at the University of Delaware in Fall, 2000. Later I added several more problems and solutions. Most of the solutions were prepared by me, but some are based on the ones given by students from the class, and from subsequent classes. I tried to acknowledge all those, and I apologize in advance if I missed someone. The course focused on Enumeration and Matching Theory. Many problems related to enumeration are taken from a 1984-85 course given by Herbert S. Wilf at the University of Pennsylvania. I would like to thank all students who took the course. Their friendly criti- cism and questions motivated some of the problems. I am especially thankful to Frank Fiedler and David Kravitz. To Frank – for sharing with me LaTeX files of his solutions and allowing me to use them. I did it several times. To David – for the technical help with the preparation of this version. Hopefully all solutions included in this document are correct, but the whole set is by no means polished. I will appreciate all comments. Please send them to [email protected]. – Felix Lazebnik 1 Problem 1. In how many 4–digit numbers abcd (a, b, c, d are the digits, a 6= 0) (i) a < b < c < d? (ii) a > b > c > d? Solution. (i) Notice that there exists a bijection between the set of our numbers and the set of all 4–subsets of the set {1, 2,..., 9}. -
Convex Geometric Connections to Information Theory
CONVEX GEOMETRIC CONNECTIONS TO INFORMATION THEORY by Justin Jenkinson Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Department of Mathematics CASE WESTERN RESERVE UNIVERSITY May, 2013 CASE WESTERN RESERVE UNIVERSITY School of Graduate Studies We hereby approve the dissertation of Justin Jenkinson, candi- date for the the degree of Doctor of Philosophy. Signed: Stanislaw Szarek Co-Chair of the Committee Elisabeth Werner Co-Chair of the Committee Elizabeth Meckes Kenneth Kowalski Date: April 4, 2013 *We also certify that written approval has been obtained for any proprietary material contained therein. c Copyright by Justin Jenkinson 2013 TABLE OF CONTENTS List of Figures . .v Acknowledgments . vi Abstract . vii Introduction . .1 CHAPTER PAGE 1 Relative Entropy of Convex Bodies . .6 1.1 Notation . .7 1.2 Background . 10 1.2.1 Affine Invariants . 11 1.2.2 Associated Bodies . 13 1.2.3 Entropy . 15 1.3 Mean Width Bodies . 16 1.4 Relative Entropies of Cone Measures and Affine Surface Areas . 24 1.5 Proof of Theorem 1.4.1 . 30 2 Geometry of Quantum States . 37 2.1 Preliminaries from Convex Geometry . 39 2.2 Summary of Volumetric Estimates for Sets of States . 44 2.3 Geometric Measures of Entanglement . 51 2.4 Ranges of Various Entanglement Measures and the role of the Di- mension . 56 2.5 Levy's Lemma and its Applications to p-Schatten Norms . 60 2.6 Concentration for the Support Functions of PPT and S ...... 64 2.7 Geometric Banach-Mazur Distance between PPT and S ...... 66 2.8 Hausdorff Distance between PPT and S in p-Schatten Metrics . -
Convex and Combinatorial Geometry Fest
Convex and Combinatorial Geometry Fest Abhinav Kumar (MIT), Daniel Pellicer (Universidad Nacional Autonoma de Mexico), Konrad Swanepoel (London School of Economics and Political Science), Asia Ivi´cWeiss (York University) February 13–February 15 2015 Karoly Bezdek and Egon Schulte are two geometers who have made important and numerous contribu- tions to Discrete Geometry. Both have trained many young mathematicians and have actively encouraged the development of the area by professors of various countries. Karoly Bezdek studied the illumination problem among many other problems about packing and covering of convex bodies. More recently he has also contributed to the theory of packing of equal balls in three-space and to the study of billiards, both of which are important for physicists. He has contributed two monographs [1, 2] in Discrete Geometry. Egon Schulte, together with Ludwig Danzer, initiated the study on abstract polytopes as a way to enclose in the same theory objects with combinatorial properties resembling those of convex polytopes. An important cornerstone on this topic is his monograph with Peter McMullen [11]. In the latest years he explored the link between abstract polytopes and crystallography. In this Workshop, which is a continuation of the preceding five-day one, we explored the ways in which the areas of abstract polytopes and discrete convex geometry has been influenced by the work of the two researchers. The last three decades have witnessed the revival of interest in these subjects and great progress has been made on many fundamental problems. 1 Abstract polytopes 1.1 Highly symmetric combinatorial structures in 3-space Asia Ivic´ Weiss kicked off the workshop, discussing joint work with Daniel Pellicer [15] on the combinatorial structure of Schulte’s chiral polyhedra. -
Convex Bodies and Algebraic Geometry an Introduction to the Theory of Toric Varieties
T. Oda Convex Bodies and Algebraic Geometry An Introduction to the Theory of Toric Varieties Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Vol. 15 The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric Softcover reprint of the original 1st ed. varieties gives new insight into continued fractions as well as their higher-dimensional 1988, VIII, 212 p. analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix. Printed book Softcover ▶ 109,99 € | £99.99 | $139.99 ▶ *117,69 € (D) | 120,99 € (A) | CHF 130.00 Order online at springer.com ▶ or for the Americas call (toll free) 1-800-SPRINGER ▶ or email us at: [email protected].