Floer Homology Groups in Yang-Mills Theory

Author: S. K. Donaldson

Publisher: Cambridge University Press

ISBN: 9781139432603

Category: Mathematics

Page: 236

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The concept of Floer homology was one of the most striking developments in differential geometry. It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles. The ideas led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory. The first half of this book gives a thorough account of Floer's construction in the context of gauge theory over 3 and 4-dimensional manifolds. The second half works out some further technical developments of the theory, and the final chapter outlines some research developments for the future - including a discussion of the appearance of modular forms in the theory. The scope of the material in this book means that it will appeal to graduate students as well as those on the frontiers of the subject.

Monopoles and Three-Manifolds

Author: Peter Kronheimer,Tomasz Mrowka

Publisher: Cambridge University Press

ISBN: 1139468669

Category: Mathematics

Page: N.A

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Originating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in three- and four-dimensional geometry and topology. This 2007 book provides a comprehensive treatment of Floer homology, based on the Seiberg–Witten monopole equations. After first providing an overview of the results, the authors develop the analytic properties of the Seiberg–Witten equations, assuming only a basic grounding in differential geometry and analysis. The Floer groups of a general three-manifold are then defined and their properties studied in detail. Two final chapters are devoted to the calculation of Floer groups and to applications of the theory in topology. Suitable for beginning graduate students and researchers, this book provides a full discussion of a central part of the study of the topology of manifolds.

A Morse-Bott Approach to Monopole Floer Homology and the Triangulation Conjecture

Author: Francesco Lin

Publisher: American Mathematical Soc.

ISBN: 1470429632

Category: Floer homology

Page: 162

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In the present work the author generalizes the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped equipped with a structure which is isomorphic to its conjugate, the author defines the counterpart in this context of Manolescu's recent Pin(2)-equivariant Seiberg-Witten-Floer homology. In particular, the author provides an alternative approach to his disproof of the celebrated Triangulation conjecture.

Floer Homology, Gauge Theory, and Low-dimensional Topology

Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfréd Rényi Institute of Mathematics, Budapest, Hungary, June 5-26, 2004

Author: Clay Mathematics Institute. Summer School

Publisher: American Mathematical Soc.

ISBN: 9780821838457

Category: Mathematics

Page: 297

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Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics, and play an important role in the description of the electro-weak and strong nuclear forces. The use of gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980s, and was revolutionized by the introduction of the Seiberg-Witten equations in the mid-1990s. Since the birth of the subject, it has retained its close connection with symplectic topology. The analogy between these two fields of study was further underscored by Andreas Floer's construction of an infinite-dimensional variant of Morse theory that applies in two a priori different contexts: either to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold, or to define topological invariants for three-manifolds, which fit into a framework for calculating invariants for smooth four-manifolds. ``Heegaard Floer homology'', the recently-discovered invariant for three- and four-manifolds, comes from an application of Lagrangian Floer homology to spaces associated to Heegaard diagrams. Although this theory is conjecturally isomorphic to Seiberg-Witten theory, it is more topological and combinatorial in flavor and thus easier to work with in certain contexts. The interaction between gauge theory, low-dimensional topology, and symplectic geometry has led to a number of striking new developments in these fields. The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a special emphasis on the very fruitful interplay between disciplines. This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute Summer School at the Alfred Renyi Institute of Mathematics in Budapest, Hungary. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from Heegaard Floer homology, contact geometry, smooth four-manifold topology, and symplectic four-manifolds.

Perspectives in Analysis, Geometry, and Topology

On the Occasion of the 60th Birthday of Oleg Viro

Author: Ilia Itenberg,Burglind Jöricke,Mikael Passare

Publisher: Springer Science & Business Media

ISBN: 0817682775

Category: Mathematics

Page: 464

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The articles in this volume are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap between analysis, geometry, and topology. The encounters between these three fields are widespread and often provide impetus for major breakthroughs in applications. Topics include new developments in low dimensional topology related to invariants of links and three and four manifolds; Perelman's spectacular proof of the Poincare conjecture; and the recent advances made in algebraic, complex, symplectic, and tropical geometry.

Modern Geometry: A Celebration of the Work of Simon Donaldson

Author: Vicente Muñoz,Ivan Smith,Richard P. Thomas

Publisher: American Mathematical Soc.

ISBN: 1470440946

Category: Four-manifolds (Topology)

Page: 416

View: 9147

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This book contains a collection of survey articles of exciting new developments in geometry, written in tribute to Simon Donaldson to celebrate his 60th birthday. Reflecting the wide range of Donaldson's interests and influence, the papers range from algebraic geometry and topology through symplectic geometry and geometric analysis to mathematical physics. Their expository nature means the book acts as an invitation to the various topics described, while also giving a sense of the links between these different areas and the unity of modern geometry.

Dynamics of Linear Operators

Author: Frédéric Bayart,Étienne Matheron

Publisher: Cambridge University Press

ISBN: 0521514967

Category: Mathematics

Page: 337

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The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.

Torsors and Rational Points

Author: Alexei Skorobogatov

Publisher: Cambridge University Press

ISBN: 9780521802376

Category: Mathematics

Page: 187

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This book, first published in 2001, is a complete and coherent exposition of the theory and applications of torsors to rational points.

Geometry & Topology

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Geometry

Page: N.A

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Fully refereed international journal dealing with all aspects of geometry and topology and their applications.

Poisson Structures

Author: Camille Laurent-Gengoux,Anne Pichereau,Pol Vanhaecke

Publisher: Springer Science & Business Media

ISBN: 3642310907

Category: Mathematics

Page: 464

View: 6327

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Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Proceedings of the International Congress of Mathematicians

Madrid, August 22-30, 2006

Author: Marta Sanz Solé

Publisher: Amer Mathematical Society

ISBN: 9783037190227

Category: Mathematics

Page: 4500

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The International Congress of Mathematicians (ICM) is held every four years. It is a major scientific event, bringing together mathematicians from all over the world and demonstrating the vital role that mathematics play in our society. In particular, the Fields Medals are awarded to recognize outstanding mathematical achievement. At the same time, the International Mathematical Union awards the Nevanlinna Prize for work in the field of theoretical computer science. The proceedings of ICM 2006, published as a three-volume set, present an overview of current research in all areas of mathematics and provide a permanent record the congress. The first volume features the works of Fields Medallists and the Nevanlinna Prize winner, the plenary lectures, and the speeches and pictures of the opening and closing ceremonies and award sessions. The other two volumes present the invited lectures, arranged according to their mathematical subject. Information for our distributors: Distributed within the Americas by the American Mathematical Society. All commerical channel discounts apply.

Geometry and Physics

XVI International Fall Workshop

Author: Rui Loja Fernandes,Roger Picken

Publisher: American Inst. of Physics

ISBN: 9780735405462

Category: Science

Page: 228

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All papers have been peer-reviewed. The XVI International Fall Workshop on Geometry and Physics brought together geometers and physicists from within and outside the Iberian peninsula, to exchange ideas on how to describe and understand a variety of phenomena in areas such as mechanics or gravity.

Tensor Categories

Author: Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik

Publisher: American Mathematical Soc.

ISBN: 1470434415

Category:

Page: 344

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Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

The Lévy Laplacian

Author: M. N. Feller

Publisher: Cambridge University Press

ISBN: 9781139447966

Category: Mathematics

Page: N.A

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The Lévy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this book was the first systematic treatment of the Lévy–Laplace operator. The book describes the infinite-dimensional analogues of finite-dimensional results, and more especially those features which appear only in the generalized context. It develops a theory of operators generated by the Lévy Laplacian and the symmetrized Lévy Laplacian, as well as a theory of linear and nonlinear equations involving it. There are many problems leading to equations with Lévy Laplacians and to Lévy–Laplace operators, for example superconductivity theory, the theory of control systems, the Gauss random field theory, and the Yang–Mills equation. The book is complemented by an exhaustive bibliography. The result is a work that will be valued by those working in functional analysis, partial differential equations and probability theory.

Algebraic Topology

Author: Allen Hatcher

Publisher: Cambridge University Press

ISBN: 9780521795401

Category: Mathematics

Page: 544

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An introductory textbook suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.

Confoliations

Author: Y. Eliashberg,William P. Thurston

Publisher: American Mathematical Soc.

ISBN: 0821807765

Category: Mathematics

Page: 66

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This book presents the first steps of a theory of confoliations designed to link geometry and topology of three-dimensional contact structures with the geometry and topology of codimension-one foliations on three-dimensional manifolds. Developing almost independently, these theories at first glance belonged to two different worlds: The theory of foliations is part of topology and dynamical systems, while contact geometry is the odd-dimensional 'brother' of symplectic geometry. However, both theories have developed a number of striking similarities. Confoliations - which interpolate between contact structures and codimension-one foliations - should help us to understand better links between the two theories. These links provide tools for transporting results from one field to the other.It's features include: a unified approach to the topology of codimension-one foliations and contact geometry; insight on the geometric nature of integrability; and, new results, in particular on the perturbation of confoliations into contact structures.

50 Years of Yang-Mills Theory

Author: G. 't Hooft

Publisher: World Scientific

ISBN: 9812567143

Category: Science

Page: 487

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On the 50th anniversary of YangOCoMills theory, this invaluable volume looks back at the developments and achievements in elementary particle physics that ensued from that beautiful idea. During the last five decades, Yang-Mills theory, which is undeniably the most important cornerstone of theoretical physics, has expanded widely. It has been investigated from many perspectives, and many new and unexpected features have been uncovered from this theory. In recent decades, apart from high energy physics, the theory has been actively applied in other branches of physics, such as statistical physics, condensed matter physics, nonlinear systems, etc. This makes the theory an indispensable topic for all who are involved in physics. An international team of experts, each of whom has left his mark on the developments of this remarkable theory, contribute essays or more detailed technical accounts to this volume. These articles highlight the new discoveries from the respective authorsOCO perspectives. The distinguished contributors are: S Adler, F A Bais, C Becchi, M Creutz, A De Rjula, B S DeWitt, F Englert, L D Faddeev, P Hasenfratz, R Jackiw, A Polyakov, V N Popov, R Stora, P van Baal, P van Nieuwenhuizen, S Weinberg, F Wilczek, E Witten, C N Yang. Included in each article are introductory and explanatory remarks by the editor, G OCOt Hooft, who is himself a major player in the development of Yang-Mills theory."

Analysis And Mathematical Physics

Author: Bullett Shaun,Fearn Tom,Smith Frank

Publisher: World Scientific

ISBN: 1786341018

Category: Mathematics

Page: 248

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This is a concise reference book on analysis and mathematical physics, leading readers from a foundation to advanced level understanding of the topic. This is the perfect text for graduate or PhD mathematical-science students looking for support in topics such as distributions, Fourier transforms and microlocal analysis, C* Algebras, value distribution of meromorphic functions, noncommutative differential geometry, differential geometry and mathematical physics, mathematical problems of general relativity, and special functions of mathematical physics. Analysis and Mathematical Physics is the sixth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Editor the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.