Perturbed Gradient Flow Trees And A Algebra Structures In Morse Cohomology

Author : Stephan Mescher
Publisher : Springer
Page : 171 pages
File Size : 41,6 Mb
Release : 2018-04-25
Category : Mathematics
ISBN : 9783319765846

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Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology by Stephan Mescher Pdf

This book elaborates on an idea put forward by M. Abouzaid on equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an A∞-algebra by means of perturbed gradient flow trajectories. This approach is a variation on K. Fukaya’s definition of Morse-A∞-categories for closed oriented manifolds involving families of Morse functions. To make A∞-structures in Morse theory accessible to a broader audience, this book provides a coherent and detailed treatment of Abouzaid’s approach, including a discussion of all relevant analytic notions and results, requiring only a basic grasp of Morse theory. In particular, no advanced algebra skills are required, and the perturbation theory for Morse trajectories is completely self-contained. In addition to its relevance for finite-dimensional Morse homology, this book may be used as a preparation for the study of Fukaya categories in symplectic geometry. It will be of interest to researchers in mathematics (geometry and topology), and to graduate students in mathematics with a basic command of the Morse theory.

Author : Anonim
Publisher : Unknown
Page : 868 pages
File Size : 44,5 Mb
Release : 2007
Category : Mathematics
ISBN : UOM:39015078588590

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Mathematical Reviews by Anonim Pdf

Author : Frédéric Bourgeois,Vincent Colin,András Stipsicz
Publisher : Springer Science & Business Media
Page : 538 pages
File Size : 44,9 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 : Danny Calegari
Publisher : Oxford University Press on Demand
Page : 378 pages
File Size : 49,6 Mb
Release : 2007-05-17
Category : Mathematics
ISBN : 9780198570080

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Foliations and the Geometry of 3-Manifolds by Danny Calegari Pdf

This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.

Author : Tamal Krishna Dey,Yusu Wang
Publisher : Cambridge University Press
Page : 455 pages
File Size : 52,7 Mb
Release : 2022-03-10
Category : Computers
ISBN : 9781009098168

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Computational Topology for Data Analysis by Tamal Krishna Dey,Yusu Wang Pdf

This book provides a computational and algorithmic foundation for techniques in topological data analysis, with examples and exercises.

Author : Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston
Publisher : American Mathematical Soc.
Page : 279 pages
File Size : 45,5 Mb
Release : 2018-08-09
Category : Floer homology
ISBN : 9781470428884

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Bordered Heegaard Floer Homology by Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston Pdf

The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ^HF of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ^HF. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

Author : Robin Pemantle,Mark C. Wilson
Publisher : Cambridge University Press
Page : 395 pages
File Size : 47,9 Mb
Release : 2013-05-31
Category : Mathematics
ISBN : 9781107031579

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Analytic Combinatorics in Several Variables by Robin Pemantle,Mark C. Wilson Pdf

Aimed at graduate students and researchers in enumerative combinatorics, this book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective.

Author : Dusa McDuff,Dietmar Salamon
Publisher : American Mathematical Soc.
Page : 220 pages
File Size : 50,9 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 the quantum multiplication exists and 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 Grassmanians, which relates to the Verlinde algebra. The Dubrovin connection, Gromov-Witten potential on quantum cohomology, and curve counting formulas are also discussed.

Author : Steve Y. Oudot
Publisher : American Mathematical Soc.
Page : 218 pages
File Size : 54,6 Mb
Release : 2017-05-17
Category : Electronic
ISBN : 9781470434434

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Persistence Theory: From Quiver Representations to Data Analysis by Steve Y. Oudot Pdf

Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work. The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.

Author : Mark Gross
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 40,8 Mb
Release : 2011-01-20
Category : Mathematics
ISBN : 9780821852323

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Tropical Geometry and Mirror Symmetry by Mark Gross Pdf

Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for ``integral tropical manifolds.'' A complete version of the argument is given in two dimensions.

Author : Y. Eliashberg,Yakov Eliashberg,Boris A. Khesin,François Lalonde
Publisher : American Mathematical Soc.
Page : 210 pages
File Size : 41,9 Mb
Release : 2003
Category : Mathematics
ISBN : 9780821831625

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Symplectic and Contact Topology: Interactions and Perspectives by Y. Eliashberg,Yakov Eliashberg,Boris A. Khesin,François Lalonde Pdf

The papers presented in this volume are written by participants of the ``Symplectic and Contact Topology, Quantum Cohomology, and Symplectic Field Theory'' symposium. The workshop was part of a semester-long joint venture of The Fields Institute in Toronto and the Centre de Recherches Mathematiques in Montreal. The twelve papers cover the following topics: Symplectic Topology, the interaction between symplectic and other geometric structures, and Differential Geometry and Topology. The Proceeding concludes with two papers that have a more algebraic character. One is related to the program of Homological Mirror Symmetry: the author defines a category of extended complex manifolds and studies its properties. The subject of the final paper is Non-commutative Symplectic Geometry, in particular the structure of the symplectomorphism group of a non-commutative complex plane. The in-depth articles make this book a useful reference for graduate students as well as research mathematicians.

Author : Dusa McDuff,Dietmar Salamon
Publisher : American Mathematical Soc.
Page : 744 pages
File Size : 44,7 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 : Michele Audin,Jacques Lafontaine
Publisher : Birkhäuser
Page : 333 pages
File Size : 45,7 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 : Terence Tao
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 52,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821848838

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Poincare's Legacies, Part I by Terence Tao Pdf

Focuses on ergodic theory, combinatorics, and number theory. This book discusses a variety of topics, ranging from developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas-Lehmer test for Mersenne primes.

Author : Mark Goresky,Robert MacPherson
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642717147

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Stratified Morse Theory by Mark Goresky,Robert MacPherson Pdf

Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.