Using The Borsuk Ulam Theorem
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| Author | : Jiri Matousek |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 221 |
| Release | : 2008-01-12 |
| Genre | : Mathematics |
| ISBN | : 3540766499 |
To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.
| Author | : Mahender Singh |
| Publisher | : Springer |
| Total Pages | : 318 |
| Release | : 2019-02-02 |
| Genre | : Mathematics |
| ISBN | : 9811357420 |
This book highlights the latest advances in algebraic topology, from homotopy theory, braid groups, configuration spaces and toric topology, to transformation groups and the adjoining area of knot theory. It consists of well-written original research papers and survey articles by subject experts, most of which were presented at the “7th East Asian Conference on Algebraic Topology” held at the Indian Institute of Science Education and Research (IISER), Mohali, Punjab, India, from December 1 to 6, 2017. Algebraic topology is a broad area of mathematics that has seen enormous developments over the past decade, and as such this book is a valuable resource for graduate students and researchers working in the field.
| Author | : Michael Starbird |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 313 |
| Release | : 2020-09-10 |
| Genre | : Education |
| ISBN | : 1470462613 |
Topology Through Inquiry is a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat. The point-set material contains many interesting topics well beyond the basic core, including continua and metrizability. Geometric and algebraic topology topics include the classification of 2-manifolds, the fundamental group, covering spaces, and homology (simplicial and singular). A unique feature of the introduction to homology is to convey a clear geometric motivation by starting with mod 2 coefficients. The authors are acknowledged masters of IBL-style teaching. This book gives students joy-filled, manageable challenges that incrementally develop their knowledge and skills. The exposition includes insightful framing of fruitful points of view as well as advice on effective thinking and learning. The text presumes only a modest level of mathematical maturity to begin, but students who work their way through this text will grow from mathematics students into mathematicians. Michael Starbird is a University of Texas Distinguished Teaching Professor of Mathematics. Among his works are two other co-authored books in the Mathematical Association of America's (MAA) Textbook series. Francis Su is the Benediktsson-Karwa Professor of Mathematics at Harvey Mudd College and a past president of the MAA. Both authors are award-winning teachers, including each having received the MAA's Haimo Award for distinguished teaching. Starbird and Su are, jointly and individually, on lifelong missions to make learning—of mathematics and beyond—joyful, effective, and available to everyone. This book invites topology students and teachers to join in the adventure.
| Author | : Thomas Bartsch |
| Publisher | : Springer |
| Total Pages | : 162 |
| Release | : 2006-11-15 |
| Genre | : Mathematics |
| ISBN | : 3540480994 |
Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in elliptic equations on symmetric domains or in the corresponding semiflows; and when one is looking for "special" solutions of these problems. This book is concerned with Lusternik-Schnirelmann theory and Morse-Conley theory for group invariant functionals. These topological methods are developed in detail with new calculations of the equivariant Lusternik-Schnirelmann category and versions of the Borsuk-Ulam theorem for very general classes of symmetry groups. The Morse-Conley theory is applied to bifurcation problems, in particular to the bifurcation of steady states and hetero-clinic orbits of O(3)-symmetric flows; and to the existence of periodic solutions nearequilibria of symmetric Hamiltonian systems. Some familiarity with the usualminimax theory and basic algebraic topology is assumed.
| Author | : Martin Loebl |
| Publisher | : Springer |
| Total Pages | : 829 |
| Release | : 2017-10-11 |
| Genre | : Computers |
| ISBN | : 3319444794 |
This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiří Matoušek’s numerous areas of mathematical interest.
| Author | : Anant R. Shastri |
| Publisher | : CRC Press |
| Total Pages | : 317 |
| Release | : 2011-03-04 |
| Genre | : Mathematics |
| ISBN | : 1439831637 |
Derived from the author's course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topol
| Author | : Victor Guillemin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 242 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821851934 |
Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.
| Author | : Larry Guth |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 287 |
| Release | : 2016-06-10 |
| Genre | : Mathematics |
| ISBN | : 1470428903 |
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
| Author | : Jakob Jonsson |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 376 |
| Release | : 2007-11-15 |
| Genre | : Mathematics |
| ISBN | : 3540758585 |
A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory.
| Author | : J. van Mill |
| Publisher | : Elsevier |
| Total Pages | : 414 |
| Release | : 1988-12-01 |
| Genre | : Mathematics |
| ISBN | : 0080933688 |
The first part of this book is a text for graduate courses in topology. In chapters 1 - 5, part of the basic material of plane topology, combinatorial topology, dimension theory and ANR theory is presented. For a student who will go on in geometric or algebraic topology this material is a prerequisite for later work. Chapter 6 is an introduction to infinite-dimensional topology; it uses for the most part geometric methods, and gets to spectacular results fairly quickly. The second part of this book, chapters 7 & 8, is part of geometric topology and is meant for the more advanced mathematician interested in manifolds. The text is self-contained for readers with a modest knowledge of general topology and linear algebra; the necessary background material is collected in chapter 1, or developed as needed.One can look upon this book as a complete and self-contained proof of Toruńczyk's Hilbert cube manifold characterization theorem: a compact ANR X is a manifold modeled on the Hilbert cube if and only if X satisfies the disjoint-cells property. In the process of proving this result several interesting and useful detours are made.
