From groups to geometry and back / Vaugh Climenhaga, Anatole Katok

By:

Climenhaga, Vaughn, 1982- [VerfasserIn]

Contributor(s):

Katok, Anatole, 1944-2018 [VerfasserIn]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Student Mathematical Library ; volume 81Publisher: Providence, Rhode Island : American Mathematical Society, [2017]Copyright date: © 2017Description: 1 Online-Ressource (442 p)ISBN:

9781470437534

Subject(s):

Additional physical formats: 9781470434793 | Erscheint auch als: From groups to geometry and back. Druck-Ausgabe Providence, Rhode Island : American Mathematical Society, 2017. xix, 420 SeitenMSC: MSC: *20-01 | 51-01 | 51M09 | 57M07RVK: RVK: SK 260 | SK 380 | SK 340LOC classification:

QA174.2

Online resources:

Zugang im Netz des KIT

Summary: 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 hySummary: Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objectsLecture 7. Isometries of \RR² and \RR³; a. Groups related to geometric objects; b. Symmetries of bodies in \RR²; c. Symmetries of bodies in \RR³; Lecture 8. Classifying isometries of \RR²; a. Isometries of the plane; b. Even and odd isometries; c. Isometries are determined by three points; d. Isometries are products of reflections; e. Isometries in \RR³; Lecture 9. The isometry group as a semidirect product; a. The group structure of \Isom(\RR²); b. \Isom⁺(\RR²) and its subgroups _{\pp}⁺ and \TTTSummary: Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objectsLecture 7. Isometries of \RR² and \RR³; a. Groups related to geometric objects; b. Symmetries of bodies in \RR²; c. Symmetries of bodies in \RR³; Lecture 8. Classifying isometries of \RR²; a. Isometries of the plane; b. Even and odd isometries; c. Isometries are determined by three points; d. Isometries are products of reflections; e. Isometries in \RR³; Lecture 9. The isometry group as a semidirect product; a. The group structure of \Isom(\RR²); b. \Isom⁺(\RR²) and its subgroups _{\pp}⁺ and \TTTSummary: Cover; Title page; Contents; Foreword: MASS at Penn State University; Preface; Guide for instructors; Chapter 1. Elements of group theory; Lecture 1. First examples of groups; a. Binary operations; b. Monoids, semigroups, and groups; c. Examples from numbers and multiplication tables; Lecture 2. More examples and definitions; a. Residues; b. Groups and arithmetic; c. Subgroups; d. Homomorphisms and isomorphisms; Lecture 3. First attempts at classification; a. Bird's-eye view; b. Cyclic groups; c. Direct products; d. Lagrange's Theorem; Lecture 4. Non-abelian groups and factor groupsSummary: Lecture 13. The rest of the story in \RR³a. Regular polyhedra; b. Completion of classification of isometries of \RR³; Lecture 14. A more algebraic approach; a. From synthetic to algebraic: Scalar products; b. Convex polytopes; c. Regular polytopes; Chapter 3. Groups of matrices: Linear algebra and symmetry in various geometries; Lecture 15. Euclidean isometries and linear algebra; a. Orthogonal matrices and isometries of \RRⁿ; b. Eigenvalues, eigenvectors, and diagonalizable matrices; c. Complexification, complex eigenvectors, and rotations; d. Differing multiplicities and Jordan blocksSummary: 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 hyPPN: PPN: 897974301Package identifier: Produktsigel: ZDB-4-NLEBK

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