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Books > Science & Mathematics > Mathematics > Topology
Sponsored by Carnegie-Mellon University and the University of
Pittsburgh
Back in print from the AMS, the first part of this book is an
introduction to the general theory of representations of algebraic
group schemes. Here, Janzten describes important basic notions:
induction functors, cohomology, quotients, Frobenius kernels, and
reduction mod $p$, among others. The second part of the book is
devoted to the representation theory of reductive algebraic groups
and includes topics such as the description of simple modules,
vanishing theorems, the Borel-Bott-Weil theorem and Weyl's
character formula, and Schubert schemes and line bundles on them.
This is a significantly revised edition of a modern classic. The
author has added nearly 150 pages of new material describing later
developments and has made major revisions to parts of the old text.
It continues to be the ultimate source of information on
representations of algebraic groups in finite characteristics. The
book is suitable for graduate students and research mathematicians
interested in algebraic groups and their representations. Algebra,
as a subdiscipline of mathematics, arguably has a history going
back some 4000 years to ancient Mesopotamia.The history, however,
of what is recognized today as high school algebra is much shorter,
extending back to the sixteenth century, while the history of what
practicing mathematicians call ""modern algebra"" is even shorter
still. The present volume provides a glimpse into the complicated
and often convoluted history of this latter conception of algebra
by juxtaposing twelve episodes in the evolution of modern algebra
from the early nineteenth-century work of Charles Babbage on
functional equations to Alexandre Grothendieck's
mid-twentieth-century metaphor of a ""rising sea"" in his
categorical approach to algebraic geometry. In addition to
considering the technical development of various aspects of
algebraic thought, the historians of modern algebra whose work is
united in this volume explore such themes as the changing aims and
organization of the subject as well as the often complex lines of
mathematical communication within and across national boundaries.
Among the specific algebraic ideas considered are the concept of
divisibility and the introduction of non-commutative algebras into
the study of number theory and the emergence of algebraic geometry
in the twentieth century.The resulting volume is essential reading
for anyone interested in the history of modern mathematics in
general and modern algebra in particular. It will be of particular
interest to mathematicians and historians of mathematics.
Information for our distributors: Co-published with the London
Mathematical Society beginning with Volume 4. Members of the LMS
may order directly from the AMS at the AMS member price. The LMS is
registered with the Charity Commissioners.
The language of -categories provides an insightful new way of
expressing many results in higher-dimensional mathematics but can
be challenging for the uninitiated. To explain what exactly an
-category is requires various technical models, raising the
question of how they might be compared. To overcome this, a
model-independent approach is desired, so that theorems proven with
any model would apply to them all. This text develops the theory of
-categories from first principles in a model-independent fashion
using the axiomatic framework of an -cosmos, the universe in which
-categories live as objects. An -cosmos is a fertile setting for
the formal category theory of -categories, and in this way the
foundational proofs in -category theory closely resemble the
classical foundations of ordinary category theory. Equipped with
exercises and appendices with background material, this first
introduction is meant for students and researchers who have a
strong foundation in classical 1-category theory.
Algebraic groups and Lie groups are important in most major areas
of mathematics, occuring in diverse roles such as the symmetries of
differential equations and as central figures in the Langlands
program for number theory. In this book, Professor Borel looks at
the development of the theory of Lie groups and algebraic groups,
highlighting the evolution from the almost purely local theory at
the start to the global theory that we know today. As the starting
point of this passage from local to global, the author takes Lie's
theory of local analytic transformation groups and Lie algebras. He
then follows the globalization of the process in its two most
important frameworks: (transcendental) differential geometry and
algebraic geometry. Chapters II to IV are devoted to the former,
Chapters V to VIII, to the latter.The essays in the first part of
the book survey various proofs of the full reducibility of linear
representations of $SL 2M$, the contributions H. Weyl to
representation and invariant theory for Lie groups, and conclude
with a chapter on E. Cartan's theory of symmetric spaces and Lie
groups in the large. The second part of the book starts with
Chapter V describing the development of the theory of linear
algebraic groups in the 19th century. Many of the main
contributions here are due to E. Study, E. Cartan, and above all,
to L. Maurer. After being abandoned for nearly 50 years, the theory
was revived by Chevalley and Kolchin and then further developed by
many others. This is the focus of Chapter VI. The book concludes
with two chapters on various aspects of the works of Chevalley on
Lie groups and algebraic groups and Kolchin on algebraic groups and
the Galois theory of differential fields. The author brings a
unique perspective to this study. As an important developer of some
of the modern elements of both the differential geometric and the
algebraic geometric sides of the theory, he has a particularly deep
appreciation of the underlying mathematics. His lifelong
involvement and his historical research in the subject give him a
special appreciation of the story of its development.
Carl Friedrich Gauss, the "foremost of mathematicians," was a land
surveyor. Measuring and calculating geodetic networks on the curved
Earth was the inspiration for some of his greatest mathematical
discoveries. This is just one example of how mathematics and
geodesy, the science and art of measuring and mapping our world,
have evolved together throughout history. This text is for students
and professionals in geodesy, land surveying, and geospatial
science who need to understand the mathematics of describing the
Earth and capturing her in maps and geospatial data: the discipline
known as mathematical geodesy. Map of the World: An Introduction to
Mathematical Geodesy aims to provide an accessible introduction to
this area, presenting and developing the mathematics relating to
maps, mapping, and the production of geospatial data. Described are
the theory and its fundamental concepts, its application for
processing, analyzing, transforming, and projecting geospatial
data, and how these are used in producing charts and atlases. Also
touched upon are the multitude of cross-overs into other sciences
sharing in the adventure of discovering what our world really looks
like. FEATURES * Written in a fluid and accessible style, replete
with exercises; adaptable for courses on different levels. *
Suitable for students and professionals in the mapping sciences,
but also for lovers of maps and map making.
Providing a succinct yet comprehensive treatment of the essentials
of modern differential geometry and topology, this book's clear
prose and informal style make it accessible to advanced
undergraduate and graduate students in mathematics and the physical
sciences. The text covers the basics of multilinear algebra,
differentiation and integration on manifolds, Lie groups and Lie
algebras, homotopy and de Rham cohomology, homology, vector
bundles, Riemannian and pseudo-Riemannian geometry, and degree
theory. It also features over 250 detailed exercises, and a variety
of applications revealing fundamental connections to classical
mechanics, electromagnetism (including circuit theory), general
relativity and gauge theory. Solutions to the problems are
available for instructors at www.cambridge.org/9781107042193.
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