Basic Noncommutative Geometry (ems Series Of Lectures In Mathematics)
by Masoud Khalkhali /
2013 / English / PDF
3.1 MB Download
This text provides an introduction to noncommutative geometry and
some of its applications. It can be used either as a textbook for a
graduate course or for self-study. It will be useful for graduate
students and researchers in mathematics and theoretical physics and
all those who are interested in gaining an understanding of the
subject. One feature of this book is the wealth of examples and
exercises that help the reader to navigate through the subject.
While background material is provided in the text and in several
appendices, some familiarity with basic notions of functional
analysis, algebraic topology, differential geometry and homological
algebra at a first year graduate level is helpful. Developed by
Alain Connes since the late 1970s, noncommutative geometry has
found many applications to long-standing conjectures in topology
and geometry and has recently made headways in theoretical physics
and number theory. The book starts with a detailed description of
some of the most pertinent algebra geometry correspondences by
casting geometric notions in algebraic terms, then proceeds in the
second chapter to the idea of a noncommutative space and how it is
constructed. The last two chapters deal with homological tools:
cyclic cohomology and Connes-Chern characters in -theory and
-homology, culminating in one commutative diagram expressing the
equality of topological and analytic index in a noncommutative
setting. Applications to integrality of noncommutative topological
invariants are given as well. Two new sections have been added to
the second edition: the first new section concerns the Gauss-Bonnet
theorem and the definition and computation of the scalar curvature
of the curved noncommutative two torus, and the second new section
is a brief introduction to Hopf cyclic cohomology. The bibliography
has been extended and some new examples are presented.
This text provides an introduction to noncommutative geometry and
some of its applications. It can be used either as a textbook for a
graduate course or for self-study. It will be useful for graduate
students and researchers in mathematics and theoretical physics and
all those who are interested in gaining an understanding of the
subject. One feature of this book is the wealth of examples and
exercises that help the reader to navigate through the subject.
While background material is provided in the text and in several
appendices, some familiarity with basic notions of functional
analysis, algebraic topology, differential geometry and homological
algebra at a first year graduate level is helpful. Developed by
Alain Connes since the late 1970s, noncommutative geometry has
found many applications to long-standing conjectures in topology
and geometry and has recently made headways in theoretical physics
and number theory. The book starts with a detailed description of
some of the most pertinent algebra geometry correspondences by
casting geometric notions in algebraic terms, then proceeds in the
second chapter to the idea of a noncommutative space and how it is
constructed. The last two chapters deal with homological tools:
cyclic cohomology and Connes-Chern characters in -theory and
-homology, culminating in one commutative diagram expressing the
equality of topological and analytic index in a noncommutative
setting. Applications to integrality of noncommutative topological
invariants are given as well. Two new sections have been added to
the second edition: the first new section concerns the Gauss-Bonnet
theorem and the definition and computation of the scalar curvature
of the curved noncommutative two torus, and the second new section
is a brief introduction to Hopf cyclic cohomology. The bibliography
has been extended and some new examples are presented.