Fukaya Categories And Picard Lefschetz Theory

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Fukaya Categories And Picard Lefschetz Theory

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Author : Paul Seidel
language : en
Publisher: European Mathematical Society
Release Date : 2008

Fukaya Categories And Picard Lefschetz Theory written by Paul Seidel and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra. Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry.

Algebra Geometry And Physics In The 21st Century

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Author : Denis Auroux
language : en
Publisher: Birkhäuser
Release Date : 2017-07-27

Algebra Geometry And Physics In The 21st Century written by Denis Auroux and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-07-27 with Mathematics categories.

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren

Homological Mirror Symmetry And Tropical Geometry

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Author : Ricardo Castano-Bernard
language : en
Publisher: Springer
Release Date : 2014-10-07

Homological Mirror Symmetry And Tropical Geometry written by Ricardo Castano-Bernard and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-07 with Mathematics categories.

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.

Contact And Symplectic Topology

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Author : Frédéric Bourgeois
language : en
Publisher: Springer Science & Business Media
Release Date : 2014-03-10

Contact And Symplectic Topology written by Frédéric Bourgeois and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-10 with Science categories.

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.

A Gentle Introduction To Homological Mirror Symmetry

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Author : Raf Bocklandt
language : en
Publisher: Cambridge University Press
Release Date : 2021-08-19

A Gentle Introduction To Homological Mirror Symmetry written by Raf Bocklandt and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-08-19 with Mathematics categories.

Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.

Lectures On Factorization Homology Categories And Topological Field Theories

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Author : Hiro Lee Tanaka
language : en
Publisher: Springer Nature
Release Date : 2020-12-14

Lectures On Factorization Homology Categories And Topological Field Theories written by Hiro Lee Tanaka and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-14 with Science categories.

This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely read suitable for advanced graduate students and interested researchers in topology and adjacent fields.

Lagrangian Intersection Floer Theory

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Author : Kenji Fukaya
language : en
Publisher: American Mathematical Soc.
Release Date : 2010-06-21

Lagrangian Intersection Floer Theory written by Kenji Fukaya and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-06-21 with Mathematics categories.

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.

From Representation Theory To Mathematical Physics And Back

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Author : Mikhail Khovanov
language : en
Publisher: American Mathematical Society
Release Date : 2025-05-14

From Representation Theory To Mathematical Physics And Back written by Mikhail Khovanov and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-14 with Mathematics categories.

This volume is a proceedings of a workshop at the Simons Center for Geometry and Physics from May 31– June 4, 2022. The workshop highlighted progress in the areas of vertex operator algebras, conformal field theory, categorification, low dimensional topology and representation theory of affine Lie algebras, loop groups, and quantum groups. In the past 40 years, string theory gave rise to the mathematical theory of vertex operator algebras, which led to the construction of representations of affine Lie algebras and the Moonshine module of the Monster group. These mathematical constructions have in turn led to ideas about 3-dimensional quantum gravity. In another direction, the discovery of the Jones polynomial led to a physical construction of 3-dimensional topological quantum field theories (TQFTs), which in turn advanced many mathematical developments in quantum groups and low dimensional topology. Louis Crane and Igor Frenkel introduced the categorification program with the goal of upgrading 3-dimensional TQFTs coming from representation theory of quantum groups to 4-dimensional TQFTs. This idea gave rise to the development of link homologies constructed from representation-theoretic, algebraic-geometric, combinatorial, and physical structures. Articles in this volume present both classical and new results related to these topics. They will be interesting to researchers and graduate students working in mathematical aspects of modern quantum field theory.

Symplectic Topology And Floer Homology

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Author : Yong-Geun Oh
language : en
Publisher: Cambridge University Press
Release Date : 2015-08-27

Symplectic Topology And Floer Homology written by Yong-Geun Oh and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-27 with Mathematics categories.

The second part of a two-volume set offering a systematic explanation of symplectic topology. This volume provides a comprehensive introduction to Hamiltonian and Lagrangian Floer theory.

Lagrangian Floer Theory And Its Deformations

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Author : Yong-Geun Oh
language : en
Publisher: Springer Nature
Release Date : 2024-06-06

Lagrangian Floer Theory And Its Deformations written by Yong-Geun Oh and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-06 with Mathematics categories.

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.