Lagrangian Intersection Floer Theory

Author : Kenji Fukaya
Publisher : American Mathematical Soc.
Page : 396 pages
File Size : 48,8 Mb
Release : 2010-06-28
Category : Floer homology
ISBN : 9780821852491

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Lagrangian Intersection Floer Theory by Kenji Fukaya Pdf

This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered $A_\infty$-algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered $A_\infty$ algebras and $A_\infty$ bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Volume II contains detailed studies of two of the main points of the foundation of the theory: transversality and orientation. The study of transversality is based on the virtual fundamental chain techniques (the theory of Kuranishi structures and their multisections) and chain level intersection theories. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. A self-contained account of the general theory of Kuranishi structures is also included in the appendix of this volume.

Author : Kenji Fukaya
Publisher : Unknown
Page : 410 pages
File Size : 46,7 Mb
Release : 2009
Category : Electronic books
ISBN : 1470417480

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Lagrangian Intersection Floer Theory by Kenji Fukaya Pdf

This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered A_\infty-algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered A_\infty algebras and A_\infty bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Vo.

Author : Kenji Fukaya
Publisher : American Mathematical Society(RI)
Page : 0 pages
File Size : 48,7 Mb
Release : 2009
Category : Floer homology
ISBN : 0821848364

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Lagrangian Intersection Floer Theory by Kenji Fukaya Pdf

Author : Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Publisher : American Mathematical Soc.
Page : 426 pages
File Size : 43,7 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9780821888490

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Lagrangian Intersection Floer Theory by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono Pdf

Author : Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 53,8 Mb
Release : 2019-09-05
Category : Floer homology
ISBN : 9781470436254

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Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory by Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono Pdf

In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating bulk deformations, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher in a slightly less general context. Then the authors explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds. The most novel part of this paper is its use of open-closed Gromov-Witten-Floer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation). The authors use this open-closed Gromov-Witten-Floer theory to produce new examples. Using the calculation of Lagrangian Floer cohomology with bulk, they produce examples of compact symplectic manifolds which admits uncountably many independent quasi-morphisms . They also obtain a new intersection result for the Lagrangian submanifold in .

Author : Yong-Geun Oh
Publisher : Springer Nature
Page : 426 pages
File Size : 48,9 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9789819717989

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Lagrangian Floer Theory and Its Deformations by Yong-Geun Oh Pdf

Author : Institute for Basic Science Center for Geometry and Physics
Publisher : Springer
Page : 0 pages
File Size : 51,8 Mb
Release : 2024-05-17
Category : Mathematics
ISBN : 9819717973

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Lagrangian Floer Theory and Its Deformations by Institute for Basic Science Center for Geometry and Physics Pdf

A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology. A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures—a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context. The book starts with a quick explanation of Stasheff polytopes and their two realizations—one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces—and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (Landau–Ginzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory.

Author : Clay Mathematics Institute. Summer School
Publisher : American Mathematical Soc.
Page : 318 pages
File Size : 51,8 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 : Vin de Silva,Joel W. Robbin,Dietmar A. Salamon
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 52,8 Mb
Release : 2014-06-05
Category : Mathematics
ISBN : 9780821898864

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Combinatorial Floer Homology by Vin de Silva,Joel W. Robbin,Dietmar A. Salamon Pdf

The authors define combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an oriented -manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology. Their proof uses a formula for the Viterbo-Maslov index for a smooth lune in a -manifold.

Author : Michèle Audin,Mihai Damian
Publisher : Springer Science & Business Media
Page : 595 pages
File Size : 54,9 Mb
Release : 2013-11-29
Category : Mathematics
ISBN : 9781447154969

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Morse Theory and Floer Homology by Michèle Audin,Mihai Damian Pdf

This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold. The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications. Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part. The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis. The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.

Author : Yong-Geun Oh
Publisher : Cambridge University Press
Page : 421 pages
File Size : 40,9 Mb
Release : 2015-08-27
Category : Mathematics
ISBN : 9781107072459

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Symplectic Topology and Floer Homology by Yong-Geun Oh Pdf

The first part of a two-volume set offering a systematic explanation of symplectic topology. This volume covers the basic materials of Hamiltonian dynamics and symplectic geometry.

Author : Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono
Publisher : Springer Nature
Page : 638 pages
File Size : 45,7 Mb
Release : 2020-10-16
Category : Mathematics
ISBN : 9789811555626

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Kuranishi Structures and Virtual Fundamental Chains by Kenji Fukaya,Yong-Geun Oh,Hiroshi Ohta,Kaoru Ono Pdf

The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.

Author : Frédéric Bourgeois,Vincent Colin,András Stipsicz
Publisher : Springer Science & Business Media
Page : 538 pages
File Size : 53,8 Mb
Release : 2014-03-10
Category : Science
ISBN : 9783319020365

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Contact and Symplectic Topology by Frédéric Bourgeois,Vincent Colin,András Stipsicz Pdf

Symplectic and contact geometry naturally emerged from the mathematical description of classical physics. The discovery of new rigidity phenomena and properties satisfied by these geometric structures launched a new research field worldwide. The intense activity of many European research groups in this field is reflected by the ESF Research Networking Programme "Contact And Symplectic Topology" (CAST). The lectures of the Summer School in Nantes (June 2011) and of the CAST Summer School in Budapest (July 2012) provide a nice panorama of many aspects of the present status of contact and symplectic topology. The notes of the minicourses offer a gentle introduction to topics which have developed in an amazing speed in the recent past. These topics include 3-dimensional and higher dimensional contact topology, Fukaya categories, asymptotically holomorphic methods in contact topology, bordered Floer homology, embedded contact homology, and flexibility results for Stein manifolds.

Author : Chungen Liu
Publisher : Springer
Page : 333 pages
File Size : 41,6 Mb
Release : 2019-05-22
Category : Mathematics
ISBN : 9789811372872

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Index theory in nonlinear analysis by Chungen Liu Pdf

This book provides detailed information on index theories and their applications, especially Maslov-type index theories and their iteration theories for non-periodic solutions of Hamiltonian systems. It focuses on two index theories: L-index theory (index theory for Lagrangian boundary conditions) and P-index theory (index theory for P-boundary conditions). In addition, the book introduces readers to recent advances in the study of index theories for symmetric periodic solutions of nonlinear Hamiltonian systems, and for selected boundary value problems involving partial differential equations.

Author : Cumrun Vafa,Shing-Tung Yau
Publisher : American Mathematical Soc.
Page : 392 pages
File Size : 55,7 Mb
Release : 2001
Category : Mathematics
ISBN : 0821821598

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Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds by Cumrun Vafa,Shing-Tung Yau Pdf

The 16 articles presented here are based on lectures given at the Winter School on Mirror Symmetry held at Harvard University in January 1999. They represent recent progress and new directions in the field. Specific topics include Floer homology and mirror symmetry, special Lagrange fibrations, special Lagrangian submanifolds, and local mirror symmetry at higher genus. Other topics include homological mirror symmetry with higher products, categorical mirror symmetry in the elliptic curve, Lagrangian torus fibration of quintic hypersurfaces, mirror symmetry and T-duality, and mirror symmetry and actions of Braid groups on derived categories. This work lacks a subject index. c. Book News Inc.