Holomorphic Curves In Low Dimensions

Author : Chris Wendl
Publisher : Springer
Page : 294 pages
File Size : 44,5 Mb
Release : 2018-06-28
Category : Mathematics
ISBN : 9783319913711

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Holomorphic Curves in Low Dimensions by Chris Wendl Pdf

This monograph provides an accessible introduction to the applications of pseudoholomorphic curves in symplectic and contact geometry, with emphasis on dimensions four and three. The first half of the book focuses on McDuff's characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. The proof presented here uses the language of Lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Emphasizing applications rather than technical results, the analytical survey mostly refers to other sources for proofs, while aiming to provide precise statements that are widely applicable, plus some informal discussion of the analytical ideas behind them. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as classifying the symplectic fillings of planar contact manifolds. This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. This book is also part of the Virtual Series on Symplectic Geometry http://www.springer.com/series/16019

Author : Chris Wendl
Publisher : Cambridge University Press
Page : 197 pages
File Size : 43,9 Mb
Release : 2020-03-26
Category : Mathematics
ISBN : 9781108497404

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Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory by Chris Wendl Pdf

An accessible introduction to the intersection theory of punctured holomorphic curves and its applications in topology.

Author : Dusa McDuff,Dietmar Salamon
Publisher : American Mathematical Soc.
Page : 744 pages
File Size : 49,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821887462

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J-holomorphic Curves and Symplectic Topology by Dusa McDuff,Dietmar Salamon Pdf

The main goal of this book is to establish the fundamental theorems of the subject in full and rigourous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associatively of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology.

Author : S. K. Donaldson,C. B. Thomas
Publisher : Cambridge University Press
Page : 277 pages
File Size : 55,7 Mb
Release : 1990
Category : Mathematics
ISBN : 9780521399784

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Geometry of Low-Dimensional Manifolds: Volume 1, Gauge Theory and Algebraic Surfaces by S. K. Donaldson,C. B. Thomas Pdf

Distinguished researchers reveal the way different subjects (topology, differential and algebraic geometry and mathematical physics) interact in a text based on LMS Durham Symposium Lectures.

Author : Olivier Collin,Stefan Friedl,Cameron Gordon,Stephan Tillmann,Liam Watson
Publisher : American Mathematical Soc.
Page : 353 pages
File Size : 44,7 Mb
Release : 2020-12-14
Category : Education
ISBN : 9781470452094

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Characters in Low-Dimensional Topology by Olivier Collin,Stefan Friedl,Cameron Gordon,Stephan Tillmann,Liam Watson Pdf

This volume contains the proceedings of a conference celebrating the work of Steven Boyer, held from June 2–6, 2018, at Université du Québec à Montréal, Montréal, Québec, Canada. Boyer's contributions to research in low-dimensional geometry and topology, and to the Canadian mathematical community, were recognized during the conference. The articles cover a broad range of topics related, but not limited, to the topology and geometry of 3-manifolds, properties of their fundamental groups and associated representation varieties.

Author : URS. VAN KOERT FRAUENFELDER (OTTO.),Otto Van Koert
Publisher : Birkhauser
Page : 388 pages
File Size : 46,5 Mb
Release : 2019-09-20
Category : Electronic
ISBN : 3030101800

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The Restricted Three-Body Problem and Holomorphic Curves by URS. VAN KOERT FRAUENFELDER (OTTO.),Otto Van Koert Pdf

The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of four-dimensional symplectizations and symplectic cobordisms, and their applications to celestial mechanics. The authors study the restricted three-body problem using recent techniques coming from the theory of pseudo-holomorphic curves. The book starts with an introduction to relevant topics in symplectic topology and Hamiltonian dynamics before introducing some well-known systems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. After an overview of different regularizations of these systems, the book continues with a discussion of periodic orbits and global surfaces of section for these and more general systems. The second half of the book is primarily dedicated to developing the theory of holomorphic curves - specifically the theory of fast finite energy planes - to elucidate the proofs of the existence results for global surfaces of section stated earlier. The book closes with a chapter summarizing the results of some numerical experiments related to finding periodic orbits and global surfaces of sections in the restricted three-body problem. This book is also part of the Virtual Series on Symplectic Geometry http: //www.springer.com/series/16019

Author : Christoph Hummel
Publisher : Birkhäuser
Page : 136 pages
File Size : 51,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034889520

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Gromov’s Compactness Theorem for Pseudo-holomorphic Curves by Christoph Hummel Pdf

This book presents the original proof of Gromov's compactness theorem for pseudo-holomorphic curves in detail. Local properties of pseudo-holomorphic curves are investigated and proved from a geometric viewpoint. Properties of particular interest are isoperimetric inequalities, a monotonicity formula, gradient bounds and the removal of singularities.

Author : Hansjörg Geiges,Kai Zehmisch
Publisher : Springer Nature
Page : 203 pages
File Size : 40,6 Mb
Release : 2023-08-07
Category : Mathematics
ISBN : 9783031360640

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A Course on Holomorphic Discs by Hansjörg Geiges,Kai Zehmisch Pdf

This textbook, based on a one-semester course taught several times by the authors, provides a self-contained, comprehensive yet concise introduction to the theory of pseudoholomorphic curves. Gromov’s nonsqueezing theorem in symplectic topology is taken as a motivating example, and a complete proof using pseudoholomorphic discs is presented. A sketch of the proof is discussed in the first chapter, with succeeding chapters guiding the reader through the details of the mathematical methods required to establish compactness, regularity, and transversality results. Concrete examples illustrate many of the more complicated concepts, and well over 100 exercises are distributed throughout the text. This approach helps the reader to gain a thorough understanding of the powerful analytical tools needed for the study of more advanced topics in symplectic topology. /divThis text can be used as the basis for a graduate course, and it is also immensely suitable for independent study. Prerequisites include complex analysis, differential topology, and basic linear functional analysis; no prior knowledge of symplectic geometry is assumed. This book is also part of the Virtual Series on Symplectic Geometry.

Author : Michele Audin,Jacques Lafontaine
Publisher : Birkhäuser
Page : 333 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034885089

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Holomorphic Curves in Symplectic Geometry by Michele Audin,Jacques Lafontaine Pdf

This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.

Author : Bahar Acu,Donatella Danielli,Marta Lewicka,Arati Pati,Saraswathy RV,Miranda Teboh-Ewungkem
Publisher : Springer Nature
Page : 364 pages
File Size : 43,7 Mb
Release : 2020-07-16
Category : Mathematics
ISBN : 9783030426873

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Advances in Mathematical Sciences by Bahar Acu,Donatella Danielli,Marta Lewicka,Arati Pati,Saraswathy RV,Miranda Teboh-Ewungkem Pdf

This volume highlights the mathematical research presented at the 2019 Association for Women in Mathematics (AWM) Research Symposium held at Rice University, April 6-7, 2019. The symposium showcased research from women across the mathematical sciences working in academia, government, and industry, as well as featured women across the career spectrum: undergraduates, graduate students, postdocs, and professionals. The book is divided into eight parts, opening with a plenary talk and followed by a combination of research paper contributions and survey papers in the different areas of mathematics represented at the symposium: algebraic combinatorics and graph theory algebraic biology commutative algebra analysis, probability, and PDEs topology applied mathematics mathematics education

Author : Clay Mathematics Institute. Summer School
Publisher : American Mathematical Soc.
Page : 318 pages
File Size : 46,7 Mb
Release : 2006
Category : Mathematics
ISBN : 0821838458

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Floer Homology, Gauge Theory, and Low-Dimensional Topology by Clay Mathematics Institute. Summer School Pdf

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 theearly 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 constructionof 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 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 tothat 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. Information for our distributors: Titles in this seriesare copublished with the Clay Mathematics Institute (Cambridge, MA).

Author : Casim Abbas,Helmut Hofer
Publisher : Springer
Page : 322 pages
File Size : 47,8 Mb
Release : 2019-03-29
Category : Mathematics
ISBN : 9783030118037

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Holomorphic Curves and Global Questions in Contact Geometry by Casim Abbas,Helmut Hofer Pdf

This book explains the foundations of holomorphic curve theory in contact geometry. By using a particular geometric problem as a starting point the authors guide the reader into the subject. As such it ideally serves as preparation and as entry point for a deeper study of the analysis underlying symplectic field theory. An introductory chapter sets the stage explaining some of the basic notions of contact geometry and the role of holomorphic curves in the field. The authors proceed to the heart of the material providing a detailed exposition about finite energy planes and periodic orbits (chapter 4) to disk filling methods and applications (chapter 9). The material is self-contained. It includes a number of technical appendices giving the geometric analysis foundations for the main results, so that one may easily follow the discussion. Graduate students as well as researchers who want to learn the basics of this fast developing theory will highly appreciate this accessible approach taken by the authors.

Author : Dusa McDuff,Dietmar Salamon
Publisher : American Mathematical Soc.
Page : 207 pages
File Size : 49,8 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821803325

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J-holomorphic Curves and Quantum Cohomology by Dusa McDuff,Dietmar Salamon Pdf

$J$-holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of $J$-holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of the existence of the Gromov-Witten invariants. The second half of the book focuses on the definition of quantum cohomology. The authors establish that this multiplication exists, and give a new proof of the Ruan-Tian result that is associative on appropriate manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, leading to quantum Chern classes and Witten's calculation for Grassmannians, which relates to the Verlinde algebra. The Dubrovin connection, Gromov-Witten potential on quantum cohomology, and curve counting formulas are also discussed. The book closes with an outline of connections to Floer theory.

Author : John Erik Fornæss,Marius Irgens,Erlend Fornæss Wold
Publisher : Springer
Page : 309 pages
File Size : 48,8 Mb
Release : 2015-11-05
Category : Mathematics
ISBN : 9783319203379

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Complex Geometry and Dynamics by John Erik Fornæss,Marius Irgens,Erlend Fornæss Wold Pdf

This book focuses on complex geometry and covers highly active topics centered around geometric problems in several complex variables and complex dynamics, written by some of the world’s leading experts in their respective fields. This book features research and expository contributions from the 2013 Abel Symposium, held at the Norwegian University of Science and Technology Trondheim on July 2-5, 2013. The purpose of the symposium was to present the state of the art on the topics, and to discuss future research directions.

Author : S. K. Donaldson,Charles Benedict Thomas
Publisher : Cambridge University Press
Page : 264 pages
File Size : 41,9 Mb
Release : 1990
Category : Low-dimensional topology
ISBN : 0521400015

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Symplectic Manifolds and Jones-Witten Theory by S. K. Donaldson,Charles Benedict Thomas Pdf