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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
Bei der Herausgabe der KLEINschen Vorlesung uber die hyper-
geometrische Funktion erschienen nur zwei Wege gangbar: Entweder
eine durchgreifende Umarbeitung, auch im grossen, oder eine
moglichst weitgehende Erhaltung der ursprunglichen Form. Vor allem
auch aus historischen Grunden wurde der letztere Weg beschritten.
Daher ist die Anordnung des Stoffes erhalten geblieben; e,s ist
nur, von kleinen Anderungen abgesehen, ein Exkurs uber homogene
Schreibweise aus der KLEINschen Vorlesung uber lineare
Differentialgleichungen ein- gefugt, ferner sind die
Schlussbemerkungen zur geometrischen Theorie im Falle komplexer
Exponenten als durch die Arbeiten von F. SCHILLING uberholt,
weggelassen. Aus dem obengenannten Grunde sind beispiels- weise
auch Entwicklungen beibehalten worden, die heute schon dem Anfanger
gelaufig sind (etwa die Ausfuhrungen uber stereographische
Projektion). In Rucksicht auf moglichste Erhaltung der KLEINschen
Darstellung sind ferner Hinweise des Herausgebers auf inzwischen
ge- machte Fortschritte der Wissenschaft vom Texte getrennt als
Anmerkun- gen am Schluss zusammengestellt. Diese Hinweise erheben
aber in keiner Weise den Anspruch auf Vollstandigkeit. Bei der
nicht zu um- gehenden Revision des Textes im einzelnen ist, dem
oben angegebenen Gesichtspunkt entsprechend, moglichste Wahrung des
personlichen KLEINschen Stils angestrebt. ubrigens habe ich darauf
Bedacht genommen, auch dem A nlanger die Lekture durch Anmerkungen
und durch Nachweise der KLEINschen Zitate zu erleichtern. Denn
zweifellos bieten gerade diese Vorlesungen eine treffliche
Erganzung und Weiterfuhrung dessen, was der Studierende mittleren
Semesters an Geometrie und Funktionentheorie kennen- gelernt hat.
The goal of this book is to provide an introduction to algebraic
geometry accessible to students. Starting from solutions of
polynomial equations, modern tools of the subject soon appear,
motivated by how they improve our understanding of geometrical
concepts. In many places, analogies and differences with related
mathematical areas are explained. The text approaches foundations
of algebraic geometry in a complete and self-contained way, also
covering the underlying algebra. The last two chapters include a
comprehensive treatment of cohomology and discuss some of its
applications in algebraic geometry.
This is essentially a book on linear algebra. But the approach is
somewhat unusual in that we emphasise throughout the geometric
aspect of the subject. The material is suitable for a course on
linear algebra for mathe matics majors at North American
Universities in their junior or senior year and at British
Universities in their second or third year. However, in view of the
structure of undergraduate courses in the United States, it is very
possible that, at many institutions, the text may be found more
suitable at the beginning graduate level. The book has two aims: to
provide a basic course in linear algebra up to, and including,
modules over a principal ideal domain; and to explain in rigorous
language the intuitively familiar concepts of euclidean, affine,
and projective geometry and the relations between them. It is
increasingly recognised that linear algebra should be approached
from a geometric point of VIew. This applies not only to
mathematics majors but also to mathematically-oriented natural
scientists and engineers."
Berkeley Lectures on p-adic Geometry presents an important
breakthrough in arithmetic geometry. In 2014, leading mathematician
Peter Scholze delivered a series of lectures at the University of
California, Berkeley, on new ideas in the theory of p-adic
geometry. Building on his discovery of perfectoid spaces, Scholze
introduced the concept of "diamonds," which are to perfectoid
spaces what algebraic spaces are to schemes. The introduction of
diamonds, along with the development of a mixed-characteristic
shtuka, set the stage for a critical advance in the discipline. In
this book, Peter Scholze and Jared Weinstein show that the moduli
space of mixed-characteristic shtukas is a diamond, raising the
possibility of using the cohomology of such spaces to attack the
Langlands conjectures for a reductive group over a p-adic field.
This book follows the informal style of the original Berkeley
lectures, with one chapter per lecture. It explores p-adic and
perfectoid spaces before laying out the newer theory of shtukas and
their moduli spaces. Points of contact with other threads of the
subject, including p-divisible groups, p-adic Hodge theory, and
Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures
on p-adic Geometry will be a useful resource for students and
scholars working in arithmetic geometry and number theory.
This book collects some surveys on current trends in discrete
mathematics and discrete geometry. The areas covered include: graph
representations, structural graphs theory, extremal graph theory,
Ramsey theory and constrained satisfaction problems.
This book presents the complete proof of the Bloch-Kato conjecture
and several related conjectures of Beilinson and Lichtenbaum in
algebraic geometry. Brought together here for the first time, these
conjectures describe the structure of etale cohomology and its
relation to motivic cohomology and Chow groups. Although the proof
relies on the work of several people, it is credited primarily to
Vladimir Voevodsky. The authors draw on a multitude of published
and unpublished sources to explain the large-scale structure of
Voevodsky's proof and introduce the key figures behind its
development. They proceed to describe the highly innovative
geometric constructions of Markus Rost, including the construction
of norm varieties, which play a crucial role in the proof. The book
then addresses symmetric powers of motives and motivic cohomology
operations. Comprehensive and self-contained, The Norm Residue
Theorem in Motivic Cohomology unites various components of the
proof that until now were scattered across many sources of varying
accessibility, often with differing hypotheses, definitions, and
language.
In the first two chapters we review the theory developped by
Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both
for Stein algebras and for the algebra of real analytic functions
on a C-analytic space. Here we find a relation between real
Nullstellensatz and seventeenth Hilbert's problem for positive
semidefinite analytic functions. Namely, a positive answer to
Hilbert's problem implies a solution for the real Nullstellensatz
more similar to the one for real polinomials. A chapter is devoted
to the state of the art on this problem that is far from a complete
answer. In the last chapter we deal with inequalities. We describe
a class of semianalytic sets defined by countably many global real
analytic functions that is stable under topological properties and
under proper holomorphic maps between Stein spaces, that is,
verifies a direct image theorem. A smaller class admits also a
decomposition into irreducible components as it happens for
semialgebraic sets. During the redaction some proofs have been
simplified with respect to the original ones.
This book is the third of a three-volume set of books on the theory
of algebras, a study that provides a consistent framework for
understanding algebraic systems, including groups, rings, modules,
semigroups and lattices. Volume I, first published in the 1980s,
built the foundations of the theory and is considered to be a
classic in this field. The long-awaited volumes II and III are now
available. Taken together, the three volumes provide a
comprehensive picture of the state of art in general algebra today,
and serve as a valuable resource for anyone working in the general
theory of algebraic systems or in related fields. The two new
volumes are arranged around six themes first introduced in Volume
I. Volume II covers the Classification of Varieties, Equational
Logic, and Rudiments of Model Theory, and Volume III covers Finite
Algebras and their Clones, Abstract Clone Theory, and the
Commutator. These topics are presented in six chapters with
independent expositions, but are linked by themes and motifs that
run through all three volumes.
This book is the second of a three-volume set of books on the
theory of algebras, a study that provides a consistent framework
for understanding algebraic systems, including groups, rings,
modules, semigroups and lattices. Volume I, first published in the
1980s, built the foundations of the theory and is considered to be
a classic in this field. The long-awaited volumes II and III are
now available. Taken together, the three volumes provide a
comprehensive picture of the state of art in general algebra today,
and serve as a valuable resource for anyone working in the general
theory of algebraic systems or in related fields. The two new
volumes are arranged around six themes first introduced in Volume
I. Volume II covers the Classification of Varieties, Equational
Logic, and Rudiments of Model Theory, and Volume III covers Finite
Algebras and their Clones, Abstract Clone Theory, and the
Commutator. These topics are presented in six chapters with
independent expositions, but are linked by themes and motifs that
run through all three volumes.
This volume consists of ten articles which provide an in-depth and
reader-friendly survey of some of the foundational aspects of
singularity theory. Authored by world experts, the various
contributions deal with both classical material and modern
developments, covering a wide range of topics which are linked to
each other in fundamental ways. Singularities are ubiquitous in
mathematics and science in general. Singularity theory interacts
energetically with the rest of mathematics, acting as a crucible
where different types of mathematical problems interact, surprising
connections are born and simple questions lead to ideas which
resonate in other parts of the subject. This is the first volume in
a series which aims to provide an accessible account of the
state-of-the-art of the subject, its frontiers, and its
interactions with other areas of research. The book is addressed to
graduate students and newcomers to the theory, as well as to
specialists who can use it as a guidebook.
This book introduces the reader to modern algebraic geometry. It
presents Grothendieck's technically demanding language of schemes
that is the basis of the most important developments in the last
fifty years within this area. A systematic treatment and motivation
of the theory is emphasized, using concrete examples to illustrate
its usefulness. Several examples from the realm of Hilbert modular
surfaces and of determinantal varieties are used methodically to
discuss the covered techniques. Thus the reader experiences that
the further development of the theory yields an ever better
understanding of these fascinating objects. The text is
complemented by many exercises that serve to check the
comprehension of the text, treat further examples, or give an
outlook on further results. The volume at hand is an introduction
to schemes. To get startet, it requires only basic knowledge in
abstract algebra and topology. Essential facts from commutative
algebra are assembled in an appendix. It will be complemented by a
second volume on the cohomology of schemes.
Continuing the theme of the previous volumes, these seminar notes
reflect general trends in the study of Geometric Aspects of
Functional Analysis, understood in a broad sense. Two classical
topics represented are the Concentration of Measure Phenomenon in
the Local Theory of Banach Spaces, which has recently had triumphs
in Random Matrix Theory, and the Central Limit Theorem, one of the
earliest examples of regularity and order in high dimensions.
Central to the text is the study of the Poincare and log-Sobolev
functional inequalities, their reverses, and other inequalities, in
which a crucial role is often played by convexity assumptions such
as Log-Concavity. The concept and properties of Entropy form an
important subject, with Bourgain's slicing problem and its variants
drawing much attention. Constructions related to Convexity Theory
are proposed and revisited, as well as inequalities that go beyond
the Brunn-Minkowski theory. One of the major current research
directions addressed is the identification of lower-dimensional
structures with remarkable properties in rather arbitrary
high-dimensional objects. In addition to functional analytic
results, connections to Computer Science and to Differential
Geometry are also discussed.
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