The Theory of Gauge Fields in Four Dimensions


Author: H. Blaine Lawson
Publisher: American Mathematical Soc.
ISBN: 0821807080
Category: Mathematics
Page: 101
View: 3095
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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.

In Search of the Riemann Zeros

Strings, Fractal Membranes and Noncommutative Spacetimes
Author: Michel Laurent Lapidus
Publisher: American Mathematical Soc.
ISBN: 9780821842225
Category: Mathematics
Page: 558
View: 7175
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Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.

Frobenius Algebras and 2-D Topological Quantum Field Theories


Author: Joachim Kock
Publisher: Cambridge University Press
ISBN: 9780521540315
Category: Mathematics
Page: 240
View: 4902
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This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.

The Geometry of Four-manifolds


Author: S. K. Donaldson,P. B. Kronheimer
Publisher: Oxford University Press
ISBN: 9780198502692
Category: Fiction
Page: 440
View: 806
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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.

The Wild World of 4-manifolds


Author: Alexandru Scorpan
Publisher: American Mathematical Soc.
ISBN: 0821837494
Category: Mathematics
Page: 609
View: 4091
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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.

On the Convergence of Some Boundary Element Methods in the Plane


Author: Keijo Ruotsalainen
Publisher: N.A
ISBN: 9789516797451
Category: Boundary element methods
Page: 26
View: 9290
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Bericht

Report
Author: Jyväskylän yliopisto. Matematiikan laitos,Osmo Pekonen
Publisher: N.A
ISBN: 9789516799608
Category: Mathematics
Page: 46
View: 4183
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Proceedings of the XVII International Conference on Differential Geometric Methods in Theoretical Physics

Chester, England 15-19 August 1988
Author: Allan I. Solomon
Publisher: World Scientific Publishing Company
ISBN: 9789971508364
Category: Mathematics
Page: 367
View: 824
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Abstracts of Papers Presented to the American Mathematical Society


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 5847
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Navier-Stokes Equations and Nonlinear Functional Analysis

Second Edition
Author: Roger Temam
Publisher: SIAM
ISBN: 0898713404
Category: Technology & Engineering
Page: 141
View: 6813
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This second edition attempts to arrive as simply as possible at some central problems in the Navier-Stokes equations.

Bulletin (new series) of the American Mathematical Society


Author: American Mathematical Society
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 2322
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Journal of the Korean Mathematical Society


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 8382
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Bulletin of the American Mathematical Society


Author: American Mathematical Society
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 4763
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Notices of the American Mathematical Society


Author: American Mathematical Society
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 1105
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Kähler metric and moduli spaces


Author: Takushirō Ochiai
Publisher: Academic Pr
ISBN: 9780120010110
Category: Mathematics
Page: 457
View: 2614
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Topology of 4-Manifolds (PMS-39)


Author: Michael H. Freedman,Frank Quinn
Publisher: Princeton University Press
ISBN: 1400861063
Category: Mathematics
Page: 268
View: 7113
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One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications, and should be accessible to most graduate students. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work includes four dimensions. It is a valuable reference for geometers and physicists who need an awareness of the topological side of the field. Originally published in 1990. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Tokyo Journal of Mathematics


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 9345
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A Splitting Formula for Spectral Flow on Closed 3-manifolds


Author: Benjamin Himpel
Publisher: N.A
ISBN: N.A
Category:
Page: 246
View: 1483
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Lectures on geometric variational problems


Author: Seiki Nishikawa,Richard M. Schoen
Publisher: Springer Verlag
ISBN: 9784431701521
Category: Mathematics
Page: 154
View: 8289
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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
ISBN: N.A
Category: Mathematics
Page: N.A
View: 483
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