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This book is the fifth and final volume of Raoul Bott's Collected
Papers. It collects all of Bott's published articles since 1991 as
well as some articles published earlier but missing in the earlier
volumes. The volume also contains interviews with Raoul Bott,
several of his previously unpublished speeches, commentaries by his
collaborators such as Alberto Cattaneo and Jonathan Weitsman on
their joint articles with Bott, Michael Atiyah's obituary of Raoul
Bott, Loring Tu's authorized biography of Raoul Bott, and
reminiscences of Raoul Bott by his friends, students, colleagues,
and collaborators, among them Stephen Smale, David Mumford, Arthur
Jaffe, Shing-Tung Yau, and Loring Tu. The mathematical articles,
many inspired by physics, encompass stable vector bundles, knot and
manifold invariants, equivariant cohomology, and loop spaces. The
nonmathematical contributions give a sense of Bott's approach to
mathematics, style, personality, zest for life, and humanity. In
one of the articles, from the vantage point of his later years,
Raoul Bott gives a tour-de-force historical account of one of his
greatest achievements, the Bott periodicity theorem. A large number
of the articles originally appeared in hard-to-find conference
proceedings or journals. This volume makes them all easily
accessible. It also features a collection of photographs giving a
panoramic view of Raoul Bott's life and his interaction with other
mathematicians.
Developed from a first-year graduate course in algebraic topology,
this text is an informal introduction to some of the main ideas of
contemporary homotopy and cohomology theory. The materials are
structured around four core areas: de Rham theory, the Cech-de Rham
complex, spectral sequences, and characteristic classes. By using
the de Rham theory of differential forms as a prototype of
cohomology, the machineries of algebraic topology are made easier
to assimilate. With its stress on concreteness, motivation, and
readability, this book is equally suitable for self-study and as a
one-semester course in topology.
Manifolds, the higher-dimensional analogs of smooth curves and
surfaces, are fundamental objects in modern mathematics. Combining
aspects of algebra, topology, and analysis, manifolds have also
been applied to classical mechanics, general relativity, and
quantum field theory. In this streamlined introduction to the
subject, the theory of manifolds is presented with the aim of
helping the reader achieve a rapid mastery of the essential topics.
By the end of the book the reader should be able to compute, at
least for simple spaces, one of the most basic topological
invariants of a manifold, its de Rham cohomology. Along the way,
the reader acquires the knowledge and skills necessary for further
study of geometry and topology. The requisite point-set topology is
included in an appendix of twenty pages; other appendices review
facts from real analysis and linear algebra. Hints and solutions
are provided to many of the exercises and problems. This work may
be used as the text for a one-semester graduate or advanced
undergraduate course, as well as by students engaged in self-study.
Requiring only minimal undergraduate prerequisites, 'Introduction
to Manifolds' is also an excellent foundation for Springer's GTM
82, 'Differential Forms in Algebraic Topology'.
This book gives a clear introductory account of equivariant
cohomology, a central topic in algebraic topology. Equivariant
cohomology is concerned with the algebraic topology of spaces with
a group action, or in other words, with symmetries of spaces. First
defined in the 1950s, it has been introduced into K-theory and
algebraic geometry, but it is in algebraic topology that the
concepts are the most transparent and the proofs are the simplest.
One of the most useful applications of equivariant cohomology is
the equivariant localization theorem of Atiyah-Bott and
Berline-Vergne, which converts the integral of an equivariant
differential form into a finite sum over the fixed point set of the
group action, providing a powerful tool for computing integrals
over a manifold. Because integrals and symmetries are ubiquitous,
equivariant cohomology has found applications in diverse areas of
mathematics and physics. Assuming readers have taken one semester
of manifold theory and a year of algebraic topology, Loring Tu
begins with the topological construction of equivariant cohomology,
then develops the theory for smooth manifolds with the aid of
differential forms. To keep the exposition simple, the equivariant
localization theorem is proven only for a circle action. An
appendix gives a proof of the equivariant de Rham theorem,
demonstrating that equivariant cohomology can be computed using
equivariant differential forms. Examples and calculations
illustrate new concepts. Exercises include hints or solutions,
making this book suitable for self-study.
This book gives a clear introductory account of equivariant
cohomology, a central topic in algebraic topology. Equivariant
cohomology is concerned with the algebraic topology of spaces with
a group action, or in other words, with symmetries of spaces. First
defined in the 1950s, it has been introduced into K-theory and
algebraic geometry, but it is in algebraic topology that the
concepts are the most transparent and the proofs are the simplest.
One of the most useful applications of equivariant cohomology is
the equivariant localization theorem of Atiyah-Bott and
Berline-Vergne, which converts the integral of an equivariant
differential form into a finite sum over the fixed point set of the
group action, providing a powerful tool for computing integrals
over a manifold. Because integrals and symmetries are ubiquitous,
equivariant cohomology has found applications in diverse areas of
mathematics and physics. Assuming readers have taken one semester
of manifold theory and a year of algebraic topology, Loring Tu
begins with the topological construction of equivariant cohomology,
then develops the theory for smooth manifolds with the aid of
differential forms. To keep the exposition simple, the equivariant
localization theorem is proven only for a circle action. An
appendix gives a proof of the equivariant de Rham theorem,
demonstrating that equivariant cohomology can be computed using
equivariant differential forms. Examples and calculations
illustrate new concepts. Exercises include hints or solutions,
making this book suitable for self-study.
Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.
This text presents a graduate-level introduction to differential
geometry for mathematics and physics students. The exposition
follows the historical development of the concepts of connection
and curvature with the goal of explaining the Chern-Weil theory of
characteristic classes on a principal bundle. Along the way we
encounter some of the high points in the history of differential
geometry, for example, Gauss' Theorema Egregium and the
Gauss-Bonnet theorem. Exercises throughout the book test the
reader's understanding of the material and sometimes illustrate
extensions of the theory. Initially, the prerequisites for the
reader include a passing familiarity with manifolds. After the
first chapter, it becomes necessary to understand and manipulate
differential forms. A knowledge of de Rham cohomology is required
for the last third of the text.Prerequisite material is contained
in author's text An Introduction to Manifolds, and can be learned
in one semester. For the benefit of the reader and to establish
common notations, Appendix A recalls the basics of manifold theory.
Additionally, in an attempt to make the exposition more
self-contained, sections on algebraic constructions such as the
tensor product and the exterior power are included. Differential
geometry, as its name implies, is the study of geometry using
differential calculus. It dates back to Newton and Leibniz in the
seventeenth century, but it was not until the nineteenth century,
with the work of Gauss on surfaces and Riemann on the curvature
tensor, that differential geometry flourished and its modern
foundation was laid. Over the past one hundred years, differential
geometry has proven indispensable to an understanding of the
physical world, in Einstein's general theory of relativity, in the
theory of gravitation, in gauge theory, and now in string theory.
Differential geometry is also useful in topology, several complex
variables, algebraic geometry, complex manifolds, and dynamical
systems, among other fields. The field has even found applications
to group theory as in Gromov's work and to probability theory as in
Diaconis's work. It is not too far-fetched to argue that
differential geometry should be in every mathematician's arsenal.
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