Gerard Venema's Books

Embeddings in Manifolds

By: Robert J. Daverman,Gerard Venema

Write a review Read reviews Take a Quiz Solve Book Puzzle

A topological embedding is a homeomorphism of one space onto a subspace of another. The book analyzes how and when objects like polyhedra or manifolds embed in a given higher-dimensional manifold. The main problem is to determine when two topological embeddings of the same object are equivalent in the sense of differing only by a homeomorphism of the ambient manifold. Knot theory is the special case of spheres smoothly embedded in spheres; in this book, much more general spaces and much more general embeddings are considered. A key aspect of the main problem is taming: when is a topological embedding of a polyhedron equivalent to a piecewise linear embedding? A central theme of the book is the fundamental role played by local homotopy properties of the complement in answering this taming question. The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimension-three proofs, including the requisite piecewise linear tools, are provided. The treatment of codimension-two embeddings includes a self-contained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimension-one embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

496

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821836978

ISBN 13

9780821836972

Exploring Advanced Euclidean Geometry with GeoGebra

By: Gerard A. Venema

Write a review Read reviews Take a Quiz Solve Book Puzzle

Exploring Advanced Euclidean Geometry with GeoGebra

By: Gerard A. Venema

This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincare disk model for hyperbolic geometry. The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

129

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1614441111

ISBN 13

9781614441113

Mutual Accompaniment as Faith-Filled Living

Recognition of the Vulnerable Other

By: Gerard J. Ryan

Write a review Read reviews Take a Quiz Solve Book Puzzle

Mutual Accompaniment as Faith-Filled Living

Recognition of the Vulnerable Other

By: Gerard J. Ryan

In this book, Gerard J. Ryan examines the interrelationship between recognition theory and theology with their respective concerns for what it means to be a human. He advocates a mutual accompaniment that reformulates recognition theory within a practical and public theology. Ryan develops this interpersonal recognition through the accompaniment of vulnerable people, particularly persons with disabilities and those who suffer from mental illness. He explores three contexts that support this mutual accompaniment and the labour of recognition. These are narrativity, the stories we live out of; vulnerability, the basic human condition common to all; and participation, the inter-relationship of humanity.

Average ratings

N/A