The Theory of Gauge Fields in Four Dimensions

Author: H. Blaine Lawson

Publisher: American Mathematical Soc.

ISBN: 0821807080

Category: Mathematics

Page: 101

View: 999

Lawson's expository lectures, presented at a CBMS Regional Conference held in Santa Barbara in August 1983, provide an indepth examination of the recent work of Simon Donaldson, and is of special interest to both geometric topologists and differential geometers. This work has excited particular interest, in light of Mike Freedman's recent profound results: the complete classification, in the simply connected case, of compact topological 4-manifolds.Arguing from deep results in gauge field theory, Donaldson has proved the nonexistence of differentiable structures on certain compact 4-manifolds. Together with Freedman's results, Donaldson's work implies the existence of exotic differentiable structures in $\mathbb R^4$-a wonderful example of the results of one mathematical discipline yielding startling consequences in another. The lectures are aimed at mature mathematicians with some training in both geometry and topology, but they do not assume any expert knowledge. In addition to a close examination of Donaldson's arguments, Lawson also presents, as background material, the foundation work in gauge theory (Uhlenbeck, Taubes, Atiyah, Hitchin, Singer, et al.) which underlies Donaldson's work.

The Geometry of Four-manifolds

Author: S. K. Donaldson,P. B. Kronheimer

Publisher: Oxford University Press

ISBN: 9780198502692

Category: Fiction

Page: 440

View: 3577

This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result have hadfar-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.

Frobenius Algebras and 2-D Topological Quantum Field Theories

Author: Joachim Kock

Publisher: Cambridge University Press

ISBN: 9780521540315

Category: Mathematics

Page: 240

View: 8259

This 2003 book connects topology and algebra using modern techniques. Numerous exercises and examples, ideal for course use.

The Wild World of 4-manifolds

Author: Alexandru Scorpan

Publisher: American Mathematical Soc.

ISBN: 0821837494

Category: Mathematics

Page: 609

View: 6931

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.

Internationale mathematische Nachrichten

Author: N.A

Publisher: N.A



Page: N.A

View: 5034

Issues for Dec. 1952- include section: Nachrichten der Österreichischen Mathematischen Gesellschaft.



Author: Jyväskylän yliopisto. Matematiikan laitos,Osmo Pekonen

Publisher: N.A

ISBN: 9789516799608

Category: Mathematics

Page: 46

View: 3369

Lectures on geometric variational problems

Author: Seiki Nishikawa,Richard M. Schoen

Publisher: Springer Verlag

ISBN: 9784431701521

Category: Mathematics

Page: 154

View: 5821

The field of geometric variational problems, that is, nonlinear problems arising in geometry and topology from the point of view of global analysis, has developed very rapidly in the last decade. It was therefore felt timely to produce a set of presentations on this subject in which leading experts would provide general survey of current research from the fundamentals to the most recent results with a view to future research. This volume will interest both mature researchers and graduate students concerned with gauge theory and low dimensional topology, theory of harmonic maps, and minimal surfaces and minimal submanifolds in Riemannian manifolds.

Mathematical Reviews

Author: N.A

Publisher: N.A


Category: Mathematics

Page: N.A

View: 2655

Das BUCH der Beweise

Author: Martin Aigner,Günter M. Ziegler

Publisher: Springer-Verlag

ISBN: 3662064545

Category: Mathematics

Page: 247

View: 2117

Die elegantesten mathematischen Beweise, spannend und für jeden Interessierten verständlich. "Der Beweis selbst, seine Ästhetik, seine Pointe geht ins Geschichtsbuch der Königin der Wissenschaften ein. Ihre Anmut offenbart sich in dem gelungenen und geschickt illustrierten Buch." Die Zeit

Nonlinear Dispersive Equations

Local and Global Analysis

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 9780821889503

Category: Mathematics

Page: 373

View: 9991

"Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.".