Smooth Four Manifolds And Complex Surfaces

Author : Robert Friedman,John W. Morgan
Publisher : Springer Science & Business Media
Page : 532 pages
File Size : 53,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662030288

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Smooth Four-Manifolds and Complex Surfaces by Robert Friedman,John W. Morgan Pdf

In 1961 Smale established the generalized Poincare Conjecture in dimensions greater than or equal to 5 [129] and proceeded to prove the h-cobordism theorem [130]. This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. By the mid 1970's the main outlines of this theory were complete, and explicit answers (especially concerning simply connected manifolds) as well as general qualitative results had been obtained. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes [131]. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver. Thus new ideas are necessary to study manifolds of these "low" dimensions.

Author : Robert Friedman,John W. Morgan
Publisher : Unknown
Page : 536 pages
File Size : 46,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662030292

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Smooth Four-Manifolds and Complex Surfaces by Robert Friedman,John W. Morgan Pdf

Author : John W. Morgan
Publisher : Princeton University Press
Page : 140 pages
File Size : 52,5 Mb
Release : 1996
Category : Mathematics
ISBN : 0691025975

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The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-manifolds by John W. Morgan Pdf

The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Author : Alexandru Scorpan
Publisher : American Mathematical Society
Page : 614 pages
File Size : 42,6 Mb
Release : 2022-01-26
Category : Mathematics
ISBN : 9781470468613

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The Wild World of 4-Manifolds by Alexandru Scorpan Pdf

What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds. —MAA Reviews The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it. — Robion C. Kirby, University of California, Berkeley This book offers a panorama of the topology of simply connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. To put things in context, the book starts with a survey of higher dimensions and of topological 4-manifolds. In the second part, the main invariant of a 4-manifold—the intersection form—and its interaction with the topology of the manifold are investigated. In the third part, as an important source of examples, complex surfaces are reviewed. In the final fourth part of the book, gauge theory is presented; this differential-geometric method has brought to light how unwieldy smooth 4-manifolds truly are, and while bringing new insights, has raised more questions than answers. The structure of the book is modular, organized into a main track of about two hundred pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.

Author : Vasile Brinzanescu
Publisher : Springer
Page : 175 pages
File Size : 44,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540498452

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Holomorphic Vector Bundles over Compact Complex Surfaces by Vasile Brinzanescu Pdf

The purpose of this book is to present the available (sometimes only partial) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic structures in a given topological vector bundle over a compact complex surface. Special features of the nonalgebraic surfaces case, like irreducible vector bundles and stability with respect to a Gauduchon metric, are considered. The reader requires a grounding in geometry at graduate student level. The book will be of interest to graduate students and researchers in complex, algebraic and differential geometry.

Author : S. K. Donaldson,P. B. Kronheimer
Publisher : Oxford University Press
Page : 464 pages
File Size : 52,7 Mb
Release : 1997
Category : Language Arts & Disciplines
ISBN : 0198502699

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The Geometry of Four-manifolds by S. K. Donaldson,P. B. Kronheimer Pdf

This text provides an accessible account to the modern study of the geometry of four-manifolds. Prerequisites are a firm grounding in differential topology and geometry, as may be gained from the first year of a graduate course.

Author : Denis Auroux,Fabrizio Catanese,Marco Manetti,Gang Tian,Paul Seidel,Ivan Smith,Bernd Siebert
Publisher : Springer Science & Business Media
Page : 363 pages
File Size : 43,7 Mb
Release : 2008-04-17
Category : Mathematics
ISBN : 9783540782780

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Symplectic 4-Manifolds and Algebraic Surfaces by Denis Auroux,Fabrizio Catanese,Marco Manetti,Gang Tian,Paul Seidel,Ivan Smith,Bernd Siebert Pdf

Modern approaches to the study of symplectic 4-manifolds and algebraic surfaces combine a wide range of techniques and sources of inspiration. Gauge theory, symplectic geometry, pseudoholomorphic curves, singularity theory, moduli spaces, braid groups, monodromy, in addition to classical topology and algebraic geometry, combine to make this one of the most vibrant and active areas of research in mathematics. It is our hope that the five lectures of the present volume given at the C.I.M.E. Summer School held in Cetraro, Italy, September 2-10, 2003 will be useful to people working in related areas of mathematics and will become standard references on these topics. The volume is a coherent exposition of an active field of current research focusing on the introduction of new methods for the study of moduli spaces of complex structures on algebraic surfaces, and for the investigation of symplectic topology in dimension 4 and higher.

Author : John W. Morgan,Kieran G. O'Grady
Publisher : Springer
Page : 231 pages
File Size : 49,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540476283

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Differential Topology of Complex Surfaces by John W. Morgan,Kieran G. O'Grady Pdf

This book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donalson's polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both thebasic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduliof sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

Author : H. Pedersen
Publisher : CRC Press
Page : 766 pages
File Size : 54,9 Mb
Release : 2021-01-07
Category : Mathematics
ISBN : 9781000110852

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Geometry and Physics by H. Pedersen Pdf

"Based on the proceedings of the Special Session on Geometry and Physics held over a six month period at the University of Aarhus, Denmark and on articles from the Summer school held at Odense University, Denmark. Offers new contributions on a host of topics that involve physics, geometry, and topology. Written by more than 50 leading international experts."

Author : John W. Morgan
Publisher : Princeton University Press
Page : 138 pages
File Size : 44,6 Mb
Release : 2014-09-08
Category : Mathematics
ISBN : 9781400865161

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The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44), Volume 44 by John W. Morgan Pdf

The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Author : Robert Friedman, John W. Morgan
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 41,9 Mb
Release : 2024-06-10
Category : Four-manifolds (Topology).
ISBN : 082188686X

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Gauge Theory and the Topology of Four-Manifolds by Robert Friedman, John W. Morgan Pdf

This text is part of the IAS/Park City Mathematics series and focuses on gauge theory and the topology of four-manifolds.

Author : Dan Abramovich,Ludmil Katzarkov
Publisher : American Mathematical Soc.
Page : 506 pages
File Size : 43,7 Mb
Release : 2009
Category : Geometry, Algebraic
ISBN : 9780821847022

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Algebraic Geometry by Dan Abramovich,Ludmil Katzarkov Pdf

This volume contains research and expository papers by some of the speakers at the 2005 AMS Summer Institute on Algebraic Geometry. Numerous papers delve into the geometry of various moduli spaces, including those of stable curves, stable maps, coherent sheaves, and abelian varieties.

Author : Donald G. Babbitt,Jane E. Kister
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 40,6 Mb
Release : 2024-06-10
Category : Mathematics
ISBN : 0821895192

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Featured Reviews in "Mathematical Reviews" 1995-1996 by Donald G. Babbitt,Jane E. Kister Pdf

This collection of reprinted 'Featured Reviews' published in Mathematical Reviews (MR) in 1995 and 1996 makes widely available informed reviews of some of the best mathematics published recently. 'Featured Reviews' were introduced in MR at the beginning of 1995 in part to provide some guidance to the current research-level literature. With the exponential growth of publications in mathematical research in the first half-century of MR, it had become essentially impossible for users of MR to identify the most important new research-level books and papers, especially in fields outside of the users' own expertise. This work identifies some of the "best" new publications, papers, and books that are expected to have a significant impact on the area of pure or applied mathematics with which researchers are concerned. All of the papers reviewed here contain interesting new ideas or applications, a deep synthesis of existing ideas, or any combination of these. The volume is intended to lead the user to important new research across all fields covered by MR.

Author : Kenji Fukaya,Dieter Kotschick,M Furuta,Toshitake Kohno
Publisher : World Scientific
Page : 266 pages
File Size : 54,6 Mb
Release : 1994-08-31
Category : Electronic
ISBN : 9789814550642

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Topology, Geometry And Field Theory - Proceedings Of The 31st International Taniguchi Symposium And Proceedings Of The Conference by Kenji Fukaya,Dieter Kotschick,M Furuta,Toshitake Kohno Pdf

Nobel Symposium 129 on Neutrino Physics was held at Haga Slott in Enköping, Sweden during August 19-24, 2004. Invited to the symposium were around 40 globally leading researchers in the field of neutrino physics, both experimental and theoretical.The dominant theme of the lectures was neutrino oscillations, which after several years were recently verified by results from the Super-Kamiokande detector in Kamioka, Japan and the SNO detector in Sudbury, Canada. Discussion focused especially on effects of neutrino oscillations derived from the presence of matter and the fact that three different neutrinos exist. Since neutrino oscillations imply that neutrinos have mass, this is the first experimental observation that fundamentally deviates from the standard model of particle physics. This is a challenge to both theoretical and experimental physics. The various oscillation parameters will be determined with increased precision in new, specially designed experiments. Theoretical physics is working intensively to insert the knowledge that neutrinos have mass into the theoretical models that describe particle physics. The lectures provided a very good description of the intensive situation in the field right now. The topics discussed also included mass models for neutrinos, neutrinos in extra dimensions as well as the “seesaw mechanism,” which provides a good description of why neutrino masses are so small.This book is A4 size and in full color.

Author : Michael Artin,Hanspeter Kraft,Reinhold Remmert
Publisher : Springer Science & Business Media
Page : 317 pages
File Size : 43,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642785023

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Duration and Change by Michael Artin,Hanspeter Kraft,Reinhold Remmert Pdf

A volume containing original essays from quite diverse fields in mathematics is something of a rarity, especially if renowned scientists show the width of their discipline to the reader. This book is just such a rarity - a veritable gem. It was written to celebrate the 50th anniversary of the mathematical research institute at Oberwolfach. The articles span a range of topics from general reflections on the place of mathematics in contemporary culture to essays dealing with aspects of algebra, analysis, geometry, coding theory, scientific computing and topology. All essays are interrelated, proving the old rule that you can divide and still conquer. A book in which every mathematician or scientist interested in mathematics will find something to take their fancy.