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An up-to-date report on the current status of important research
topics in algebraic geometry and its applications, such as
computational algebra and geometry, singularity theory algorithms,
numerical solutions of polynomial systems, coding theory,
communication networks, and computer vision. Contributions on more
fundamental aspects of algebraic geometry include expositions
related to counting points on varieties over finite fields, Mori
theory, linear systems, Abelian varieties, vector bundles on
singular curves, degenerations of surfaces, and mirror symmetry of
Calabi-Yau manifolds.
The present volume contains Friedrich Hirzebruch's works from 1987
until 2012. It is the continuation of the two volumes "Friedrich
Hirzebruch, Gesammelte Abhandlungen", published by Springer-Verlag
in 1987. The volume, edited by Joachim Schwermer, Silke
Wimmer-Zagier and Don Zagier, includes all of Friedrich
Hirzebruch's mathematical publications from this period as well as
two lecture reports written by him. These are supplemented by a
number of articles and addresses containing historical or
biographical material, as well as obituaries or appreciations of
people who were mathematically or personally close to him.
Friedrich Hirzebruch (1927 -2012) was a German mathematician,
working in the fields of topology, complex manifolds and algebraic
geometry, and a leading figure of his generation. Hirzebruch's
first great mathematical achievement was the proof, in 1954, of the
generalization of the classical Riemann-Roch theorem to higher
dimensional complex manifolds, now known as the
Hirzebruch-Riemann-Roch theorem. This used the new techniques of
sheaf cohomology and was one of the centerpieces of the explosion
of new results in geometry and topology during the 1950s. Further
generalization of this led to the Grothendieck-Riemann-Roch
theorem, and the Atiyah-Singer index theorem. He received many
awards and honors, including the Wolf prize in 1988, the
Lobachevsky prize in 1990, and fifteen honorary doctorates. These
two volumes collect the majority of his research papers, which
cover a variety of topics.
In zwei Banden sind fast alle Veroffentlichungen enthalten, die
F. Hirzebruch verfasst hat."
Friedrich Hirzebruch (1927 2012) was a German mathematician,
working in the fields of topology, complex manifolds and algebraic
geometry, and a leading figure of his generation. Hirzebruch s
first great mathematical achievement was the proof, in 1954, of the
generalization of the classical Riemann-Roch theorem to higher
dimensional complex manifolds, now known as the
Hirzebruch-Riemann-Roch theorem. This used the new techniques of
sheaf cohomology and was one of the centerpieces of the explosion
of new results in geometry and topology during the 1950s. Further
generalization of this led to the Grothendieck-Riemann-Roch
theorem, and the Atiyah-Singer index theorem. He received many
awards and honors, including the Wolf prize in 1988, the
Lobachevsky prize in 1990, and fifteen honorary doctorates. These
two volumes collect the majority of his research papers, which
cover a variety of topics."
This book consists almost entirely of papers delivered at the
Seminar on partial differential equations held at
Max-Planck-Institut in the spring of 1984. They give an insight
into important recent research activities. Some further
developments are also included.
Mathematics has a certain mystique, for it is pure and ex- act, yet
demands remarkable creativity. This reputation is reinforced by its
characteristic abstraction and its own in- dividual language, which
often disguise its origins in and connections with the physical
world. Publishing mathematics, therefore, requires special effort
and talent. Heinz G-tze, who has dedicated his life to scientific
pu- blishing, took up this challenge with his typical enthusi- asm.
This Festschrift celebrates his invaluable contribu- tions to the
mathematical community, many of whose leading members he counts
among his personal friends. The articles, written by mathematicians
from around the world and coming from diverse fields, portray the
important role of mathematics in our culture. Here, the reflections
of important mathematicians, often focused on the history of
mathematics, are collected, in recognition of Heinz G-tze's
life-longsupport of mathematics.
An up-to-date report on the current status of important research
topics in algebraic geometry and its applications, such as
computational algebra and geometry, singularity theory algorithms,
numerical solutions of polynomial systems, coding theory,
communication networks, and computer vision. Contributions on more
fundamental aspects of algebraic geometry include expositions
related to counting points on varieties over finite fields, Mori
theory, linear systems, Abelian varieties, vector bundles on
singular curves, degenerations of surfaces, and mirror symmetry of
Calabi-Yau manifolds.
This is a book about numbers - all kinds of numbers, from integers to p-adics, from rationals to octonions, from reals to infinitesimals. Who first used the standard notation for Â? Why was Hamilton obsessed with quaternions? What was the prospect for "quaternionic analysis" in the 19th century? This is the story about one of the major threads of mathematics over thousands of years. It is a story that will give the reader both a glimpse of the mystery surrounding imaginary numbers in the 17th century and also a view of some major developments in the 20th.
In recent years new topological methods, especially the theory of
sheaves founded by J. LERAY, have been applied successfully to
algebraic geometry and to the theory of functions of several
complex variables. H. CARTAN and J. -P. SERRE have shown how
fundamental theorems on holomorphically complete manifolds (STEIN
manifolds) can be for mulated in terms of sheaf theory. These
theorems imply many facts of function theory because the domains of
holomorphy are holomorphically complete. They can also be applied
to algebraic geometry because the complement of a hyperplane
section of an algebraic manifold is holo morphically complete. J.
-P. SERRE has obtained important results on algebraic manifolds by
these and other methods. Recently many of his results have been
proved for algebraic varieties defined over a field of arbitrary
characteristic. K. KODAIRA and D. C. SPENCER have also applied
sheaf theory to algebraic geometry with great success. Their
methods differ from those of SERRE in that they use techniques from
differential geometry (harmonic integrals etc. ) but do not make
any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D.
HODGE have dealt successfully with problems on integrals of the
second kind on algebraic manifolds with the help of sheaf theory. I
was able to work together with K. KODAIRA and D. C. SPENCER during
a stay at the Institute for Advanced Study at Princeton from 1952
to 1954."
During the winter term 1987/88 I gave a course at the University of
Bonn under the title "Manifolds and Modular Forms." Iwanted to
develop the theory of "Elliptic Genera" and to leam it myself on
this occasion. This theory due to Ochanine, Landweber, Stong and
others was relatively new at the time. The word "genus" is meant in
the sense of my book "Neue Topologische Methoden in der
Algebraischen Geometrie" published in 1956: A genus is a
homomorphism of the Thom cobordism ring of oriented compact
manifolds into the complex numbers. Fundamental examples are the
signature and the A-genus. The A-genus equals the arithmetic genus
of an algebraic manifold, provided the first Chem class of the
manifold vanishes. According to Atiyah and Singer it is the index
of the Dirac operator on a compact Riemannian manifold with spin
structure. The elliptic genera depend on a parameter. For special
values of the parameter one obtains the signature and the A-genus.
Indeed, the universal elliptic genus can be regarded as a modular
form with respect to the subgroup r (2) of the modular group; the
two cusps o giving the signature and the A-genus. Witten and other
physicists have given motivations for the elliptic genus by
theoretical physics using the free loop space of a manifold.
Five papers by distinguished American and European mathematicians
describe some current trends in mathematics in the perspective of
the recent past and in terms of expectations for the future. Among
the subjects discussed are algebraic groups, quadratic forms,
topological aspects of global analysis, variants of the index
theorem, and partial differential equations.
Zum Anlass des 100. Geburtstages der Deutschen
Mathematiker-Vereinigung erscheint diese Festschrift, bestehend aus
neunzehn Beitragen, in denen anerkannte Fachwissenschaftler die
Entwicklung ihres jeweiligen mathematischen Fachgebietes
beschreiben und dabei auch kritische Ruckschau auf die Geschichte
der Deutschen Mathematiker-Vereinigung seit ihrer Grundung 1890
halten. Insbesondere der erste Beitrag setzt sich intensiv mit der
Historie der Mathematik und der Mathematiker im Dritten Reich
auseinander."Mit diesem Band wird ein wichtiger Beitrag zur bisher
wenig entwickelten Geschichtsschreibung der neueren Mathematik
geleistet. (R. Siegmund-Schultze in "Deutsche Literatur-Zeitung"
1,2/1992, Bd. 113)
Die Schwierigkeit Mathematik zu lernen und zu lehren ist jedem
bekannt, der einmal mit diesem Fach in Beruhrung gekommen ist.
Begriffe wie "reelle oder komplexe Zahlen, Pi" sind zwar jedem
gelaufig, aber nur wenige wissen, was sich wirklich dahinter
verbirgt. Die Autoren dieses Bandes geben jedem, der mehr wissen
will als nur die Hulle der Begriffe, eine meisterhafte Einfuhrung
in die Magie der Mathematik und schlagen einzigartige Brucken fur
Studenten.
Die Rezensenten der ersten beiden Auflagen uberschlugen
sich."
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