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Author: Robert H. Gilman
Publisher: American Mathematical Soc.
ISBN: 0821810537
Category : Computers
Languages : en
Pages : 150
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Book Description
This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Geometric Group Theory and Computer Science held at Mount Holyoke College (South Hadley, MA). The conference was devoted to computational aspects of geometric group theory, a relatively young area of research which has grown out of an influx of ideas from topology and computer science into combinatorial group theory. The book reflects recent progress in this interesting new field. Included are articles about insights from computer experiments, applications of formal language theory, decision problems, and complexity problems. There is also a survey of open questions in combinatorial group theory. The volume will interest group theorists, topologists, and experts in automata and language theory.
Author: Robert H. Gilman
Publisher: American Mathematical Soc.
ISBN: 0821810537
Category : Computers
Languages : en
Pages : 150
Get Book Here
Book Description
This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Geometric Group Theory and Computer Science held at Mount Holyoke College (South Hadley, MA). The conference was devoted to computational aspects of geometric group theory, a relatively young area of research which has grown out of an influx of ideas from topology and computer science into combinatorial group theory. The book reflects recent progress in this interesting new field. Included are articles about insights from computer experiments, applications of formal language theory, decision problems, and complexity problems. There is also a survey of open questions in combinatorial group theory. The volume will interest group theorists, topologists, and experts in automata and language theory.
Author: P. M. Neumann
Publisher: Oxford University Press, USA
ISBN: 9780198534518
Category : Language Arts & Disciplines
Languages : en
Pages : 268
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Book Description
Contains the Oxford Mathematical Institute notes for undergraduate and first-year postgraduates. The first half of the book covers groups, the second half covers geometry and both parts contain a number of exercises.
Author: Larry C. Grove
Publisher: John Wiley & Sons
ISBN: 1118030931
Category : Mathematics
Languages : en
Pages : 228
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Book Description
An authoritative, full-year course on both group theory and ordinary character theory--essential tools for mathematics and the physical sciences One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups. Challenging yet accessible, Groups and Characters features: * An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups * A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics * Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index * Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.
Author: Roger C. Lyndon
Publisher: Cambridge University Press
ISBN: 0521316944
Category : Mathematics
Languages : en
Pages : 231
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Book Description
This 1985 book is an introduction to certain central ideas in group theory and geometry. Professor Lyndon emphasises and exploits the well-known connections between the two subjects and leads the reader to the frontiers of current research at the time of publication.
Author: Vaughn Climenhaga
Publisher: American Mathematical Soc.
ISBN: 1470434792
Category : Mathematics
Languages : en
Pages : 442
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Book Description
Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.
Author: R. P. Burn
Publisher: Cambridge University Press
ISBN: 9780521347938
Category : Mathematics
Languages : en
Pages : 260
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Book Description
Following the same successful approach as Dr. Burn's previous book on number theory, this text consists of a carefully constructed sequence of questions that will enable the reader, through participation, to study all the group theory covered by a conventional first university course. An introduction to vector spaces, leading to the study of linear groups, and an introduction to complex numbers, leading to the study of Möbius transformations and stereographic projection, are also included. Quaternions and their relationships to 3-dimensional isometries are covered, and the climax of the book is a study of the crystallographic groups, with a complete analysis of these groups in two dimensions.
Author: Michael Davis
Publisher: Princeton University Press
ISBN: 0691131384
Category : Mathematics
Languages : en
Pages : 601
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Book Description
The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
Author: Alan F. Beardon
Publisher: Springer Science & Business Media
ISBN: 1461211468
Category : Mathematics
Languages : en
Pages : 350
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Book Description
This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.
Author: Martin W. Liebeck
Publisher: Cambridge University Press
ISBN: 0521406854
Category : Mathematics
Languages : en
Pages : 505
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Book Description
This volume contains a collection of papers on the subject of the classification of finite simple groups.
Author: Jean Gallier
Publisher: Springer Nature
ISBN: 3030460401
Category : Mathematics
Languages : en
Pages : 777
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Book Description
This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications. Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry. Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics. Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.
