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What do the classification of algebraic surfaces, Weyl's dimension
formula and maximal orders in central simple algebras have in
common? All are related to a type of manifold called locally mixed
symmetric spaces in this book. The presentation emphasizes
geometric concepts and relations and gives each reader the "roter
Faden", starting from the basics and proceeding towards quite
advanced topics which lie at the intersection of differential and
algebraic geometry, algebra and topology. Avoiding technicalities
and assuming only a working knowledge of real Lie groups, the text
provides a wealth of examples of symmetric spaces. The last two
chapters deal with one particular case (Kuga fiber spaces) and a
generalization (elliptic surfaces), both of which require some
knowledge of algebraic geometry. Of interest to topologists,
differential or algebraic geometers working in areas related to
arithmetic groups, the book also offers an introduction to the
ideas for non-experts.
The book discusses a series of higher-dimensional moduli spaces, of
abelian varieties, cubic and K3 surfaces, which have embeddings in
projective spaces as very special algebraic varieties. Many of
these were known classically, but in the last chapter a new such
variety, a quintic fourfold, is introduced and studied. The text
will be of interest to all involved in the study of moduli spaces
with symmetries, and contains in addition a wealth of material
which has been only accessible in very old sources, including a
detailed presentation of the solution of the equation of 27th
degree for the lines on a cubic surface.
What do the classification of algebraic surfaces, Weyl's dimension
formula and maximal orders in central simple algebras have in
common? All are related to a type of manifold called locally mixed
symmetric spaces in this book. The presentation emphasizes
geometric concepts and relations and gives each reader the "roter
Faden", starting from the basics and proceeding towards quite
advanced topics which lie at the intersection of differential and
algebraic geometry, algebra and topology. Avoiding technicalities
and assuming only a working knowledge of real Lie groups, the text
provides a wealth of examples of symmetric spaces. The last two
chapters deal with one particular case (Kuga fiber spaces) and a
generalization (elliptic surfaces), both of which require some
knowledge of algebraic geometry. Of interest to topologists,
differential or algebraic geometers working in areas related to
arithmetic groups, the book also offers an introduction to the
ideas for non-experts.
The Oxford Users' Guide to Mathematics is one of the leading
handbooks on mathematics available. It presents a comprehensive
modern picture of mathematics and emphasises the relations between
the different branches of mathematics, and the applications of
mathematics in engineering and the natural sciences. The Oxford
User's Guide covers a broad spectrum of mathematics starting with
the basic material and progressing on to more advanced topics that
have come to the fore in the last few decades. The book is
organised into mathematical sub-disciplines including analysis,
algebra, geometry, foundations of mathematics, calculus of
variations and optimisation, theory of probability and mathematical
statistics, numerical mathematics and scientific computing, and
history of mathematics. The book is supplemented by numerous tables
on infinite series, special functions, integrals, integral
transformations, mathematical statistics, and fundamental constants
in physics. It also includes a comprehensive bibliography of key
contemporary literature as well as an extensive glossary and index.
The wealth of material, reaching across all levels and numerous
sub-disciplines, makes The Oxford User's Guide to Mathematics an
invaluable reference source for students of engineering,
mathematics, computer science, and the natural sciences, as well as
teachers, practitioners, and researchers in industry and academia.
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