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Algebra: Chapter 0 is a self-contained introduction to the main
topics of algebra, suitable for a first sequence on the subject at
the beginning graduate or upper undergraduate level. The primary
distinguishing feature of the book, compared to standard textbooks
in algebra, is the early introduction of categories, used as a
unifying theme in the presentation of the main topics. A second
feature consists of an emphasis on homological algebra: basic
notions on complexes are presented as soon as modules have been
introduced, and an extensive last chapter on homological algebra
can form the basis for a follow-up introductory course on the
subject. Approximately 1,000 exercises both provide adequate
practice to consolidate the understanding of the main body of the
text and offer the opportunity to explore many other topics,
including applications to number theory and algebraic geometry.
This will allow instructors to adapt the textbook to their specific
choice of topics and provide the independent reader with a richer
exposure to algebra. Many exercises include substantial hints, and
navigation of the topics is facilitated by an extensive index and
by hundreds of cross-references.
From rings to modules to groups to fields, this undergraduate
introduction to abstract algebra follows an unconventional path.
The text emphasizes a modern perspective on the subject, with
gentle mentions of the unifying categorical principles underlying
the various constructions and the role of universal properties. A
key feature is the treatment of modules, including a proof of the
classification theorem for finitely generated modules over
Euclidean domains. Noetherian modules and some of the language of
exact complexes are introduced. In addition, standard topics - such
as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow
Theorems, simplicity of alternating groups, standard results on
field extensions, and the Fundamental Theorem of Galois Theory -
are all treated in detail. Students will appreciate the text's
conversational style, 400+ exercises, an appendix with complete
solutions to around 150 of the main text problems, and an appendix
with general background on basic logic and naive set theory.
Written to honor the 80th birthday of William Fulton, the articles
collected in this volume (the first of a pair) present substantial
contributions to algebraic geometry and related fields, with an
emphasis on combinatorial algebraic geometry and intersection
theory. Featured topics include commutative algebra, moduli spaces,
quantum cohomology, representation theory, Schubert calculus, and
toric and tropical geometry. The range of these contributions is a
testament to the breadth and depth of Fulton's mathematical
influence. The authors are all internationally recognized experts,
and include well-established researchers as well as rising stars of
a new generation of mathematicians. The text aims to stimulate
progress and provide inspiration to graduate students and
researchers in the field.
Written to honor the 80th birthday of William Fulton, the articles
collected in this volume (the second of a pair) present substantial
contributions to algebraic geometry and related fields, with an
emphasis on combinatorial algebraic geometry and intersection
theory. Featured include commutative algebra, moduli spaces,
quantum cohomology, representation theory, Schubert calculus, and
toric and tropical geometry. The range of these contributions is a
testament to the breadth and depth of Fulton's mathematical
influence. The authors are all internationally recognized experts,
and include well-established researchers as well as rising stars of
a new generation of mathematicians. The text aims to stimulate
progress and provide inspiration to graduate students and
researchers in the field.
These are transcripts of notes taken at (some of) the lectures
given at the Mittag-Leffler Institute during the first semester of
the year 1996/97Â on Enumerative geometry and its interaction
with theoretical physics. The first part of this collection
consists of notes from talks on the basics of quantum cohomology;
the second part treats more advanced topics in quantum cohomology;
the third part consists of background material and related topics;
an appendix gives a description of Kresch's C-program Farsta for
quantum cohomology computations. These notes are meant as a series
of snapshots of quantum cohomology as seen by the speakers at the
time of their lectures. The reader should bear in mind that quantum
cohomology is a growing and rapidly changing field. Many of the
writeups have been left in the form of the original talks, which
were usually more concerned with giving motivations and a point of
view, rather than conveying detailed proofs or attempting to survey
the considerably extensive literature on the subject.
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