id: 06563831
dt: b
an: 2016e.00750
au: Hilgert, Joachim
ti: Mathematical structures. From linear algebra over rings to geometry with
sheaves. (Mathematische Strukturen. Von der linearen Algebra über
Ringen zur Geometrie mit Garben.)
so: Heidelberg: Springer Spektrum (ISBN 978-3-662-48869-0/pbk;
978-3-662-48870-6/ebook). x, 303~p. (2016).
py: 2016
pu: Heidelberg: Springer Spektrum
la: DE
cc: H15 G95
ut: algebraic structures; rings; modules; multilinear algebra; sheaves;
manifolds; algebraic varieties; algebraic schemes
ci:
li: doi:10.1007/978-3-662-48870-6
ab: The book under review is geared toward upper-level undergraduate students
who are familiar with the basics of real analysis and linear algebra.
Its main goal is to provide both a panoramic overview and a profound
introduction concerning a number of modern, more advanced mathematical
concepts permeating contemporay mathematics as a whole. In this regard,
the presentation of the interrelation between various mathematical
disciplines is particularly emphasized, in the course of which the
discussion of several fundamental mathematical structures serves as the
guiding methodological principle. As for the precise contents, the book
consists of three parts, each of which is divided into several chapters
and subsections. Part I is titled “Algebraic structures” and
contains four chapters. Chapter 1 is devoted to basic ring theory,
whereas chapter 2 discusses modules over a ring,some of their
fundamental structural properties, and concrete applications of the
latter to linear mappings, including the Jordan normal form of
matrices. Chapter 3 develops the principles of multilinear algebra for
modules over a ring, with the focus on tensor products and their
universal properties, tensor algebras, symmetric algebras, and exterior
algebras. Chapter 4 explains how the concrete algebraic structures in
chapters 2 and 3 are reflected in the more general conceptual framework
of universal algebra and category theory, with a special view toward
limits, colimits, and adjoint functors. Part II is superscribed
“Local structures” and contains the subsequent three chapters.
Chapter 5 provides an introduction to the ubiquitous toolkit of sheaf
theory, both from the categorical and from the topological point of
view. This includes étale spaces, ringed spaces, and sheaves of
modules, thereby offering a glimpse of modern algebraic geometry along
the way. Chapter 6 turns to special topological structures, more
precisely to differentiable manifolds, tangent and tensor bundles,
differential forms, integration theory on real manifolds, and the
rudiments of complex analytic functions of one variable. Finally,
chapter 7 returns to algebraic geometry by discussing algebraic sets,
algebraic varieties, and algebraic schemes in greater detail. Part III
offers an outlook to some concepts obtained by combining different
types of structures. Its only chapter (Chapter 8) is titled
“Additional structures” and touches upon Riemannian manifolds,
symplectic manifolds, Kähler manifolds, and Poisson structures.
Furthermore, affine connections on manifolds, differential fiber
bundles, and group objects in categories serve as supplementing,
illustrating examples in this context. All together, this is a very
useful book for students seeking orientation for their further
specialization in mathematics. The presentation of the material is
utmost lucid, sufficiently detailed, versatile, and didactically
refined. As such, this excellent primer is a perfect source for
further, more detailed reading, and a highly useful companion for
students in general.
rv: Werner Kleinert (Berlin)