Francis Borceux's Books

A Differential Approach to Geometry

Geometric Trilogy III

By: Francis Borceux

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A Differential Approach to Geometry

Geometric Trilogy III

By: Francis Borceux

This book presents the classical theory of curves in the plane and three-dimensional space, and the classical theory of surfaces in three-dimensional space. It pays particular attention to the historical development of the theory and the preliminary approaches that support contemporary geometrical notions. It includes a chapter that lists a very wide scope of plane curves and their properties. The book approaches the threshold of algebraic topology, providing an integrated presentation fully accessible to undergraduate-level students. At the end of the 17th century, Newton and Leibniz developed differential calculus, thus making available the very wide range of differentiable functions, not just those constructed from polynomials. During the 18th century, Euler applied these ideas to establish what is still today the classical theory of most general curves and surfaces, largely used in engineering. Enter this fascinating world through amazing theorems and a wide supply of surprising examples. Reach the doors of algebraic topology by discovering just how an integer (= the Euler-Poincaré characteristics) associated with a surface gives you a lot of interesting information on the shape of the surface. And penetrate the intriguing world of Riemannian geometry, the geometry that underlies the theory of relativity. The book is of interest to all those who teach classical differential geometry up to quite an advanced level. The chapter on Riemannian geometry is of great interest to those who have to “intuitively” introduce students to the highly technical nature of this branch of mathematics, in particular when preparing students for courses on relativity.

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Page Count

462

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3319017365

ISBN 13

9783319017365

Galois Theories

By: Francis Borceux,George Janelidze

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Galois Theories

By: Francis Borceux,George Janelidze

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, the authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience: the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. The first chapters are accessible to advanced undergraduates, with later ones at a graduate level. For all algebraists and category theorists this book will be a rewarding read.

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Page Count

360

Publisher

Cambridge University Press

Published Date

ISBN 10

0521803098

ISBN 13

9780521803090

Mal'cev, Protomodular, Homological and Semi-Abelian Categories

By: Francis Borceux,Dominique Bourn

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Mal'cev, Protomodular, Homological and Semi-Abelian Categories

By: Francis Borceux,Dominique Bourn

The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory. The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment. The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

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Page Count

504

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1402019610

ISBN 13

9781402019616

Handbook of Categorical Algebra: Volume 1, Basic Category Theory

By: Francis Borceux

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Handbook of Categorical Algebra: Volume 1, Basic Category Theory

By: Francis Borceux

The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. In particular, Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory.

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363

Publisher

Cambridge University Press

Published Date

ISBN 10

0521441781

ISBN 13

9780521441780

An Algebraic Approach to Geometry

Geometric Trilogy II

By: Francis Borceux

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An Algebraic Approach to Geometry

Geometric Trilogy II

By: Francis Borceux

This is a unified treatment of the various algebraic approaches to geometric spaces. The study of algebraic curves in the complex projective plane is the natural link between linear geometry at an undergraduate level and algebraic geometry at a graduate level, and it is also an important topic in geometric applications, such as cryptography. 380 years ago, the work of Fermat and Descartes led us to study geometric problems using coordinates and equations. Today, this is the most popular way of handling geometrical problems. Linear algebra provides an efficient tool for studying all the first degree (lines, planes) and second degree (ellipses, hyperboloids) geometric figures, in the affine, the Euclidean, the Hermitian and the projective contexts. But recent applications of mathematics, like cryptography, need these notions not only in real or complex cases, but also in more general settings, like in spaces constructed on finite fields. And of course, why not also turn our attention to geometric figures of higher degrees? Besides all the linear aspects of geometry in their most general setting, this book also describes useful algebraic tools for studying curves of arbitrary degree and investigates results as advanced as the Bezout theorem, the Cramer paradox, topological group of a cubic, rational curves etc. Hence the book is of interest for all those who have to teach or study linear geometry: affine, Euclidean, Hermitian, projective; it is also of great interest to those who do not want to restrict themselves to the undergraduate level of geometric figures of degree one or two.

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Page Count

440

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3319017330

ISBN 13

9783319017334

Twistor Theory for Riemannian Symmetric Spaces

With Applications to Harmonic Maps of Riemann Surfaces

By: Francis E. Burstall,John H. Rawnsley

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Twistor Theory for Riemannian Symmetric Spaces

With Applications to Harmonic Maps of Riemann Surfaces

By: Francis E. Burstall,John H. Rawnsley

In this monograph on twistor theory and its applications to harmonic map theory, a central theme is the interplay between the complex homogeneous geometry of flag manifolds and the real homogeneous geometry of symmetric spaces. In particular, flag manifolds are shown to arise as twistor spaces of Riemannian symmetric spaces. Applications of this theory include a complete classification of stable harmonic 2-spheres in Riemannian symmetric spaces and a Bäcklund transform for harmonic 2-spheres in Lie groups which, in many cases, provides a factorisation theorem for such spheres as well as gap phenomena. The main methods used are those of homogeneous geometry and Lie theory together with some algebraic geometry of Riemann surfaces. The work addresses differential geometers, especially those with interests in minimal surfaces and homogeneous manifolds.

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