Overview
- Explains all mathematical ingredients that provide access to the complex theory of Lagrangian intersection homology
- Offers a reader-friendly introduction to the relevant homological algebra of filtered A-infinity algebras
- Starts with a quick explanation of Stasheff polytopes and their two realizations
Part of the book series: KIAS Springer Series in Mathematics (KIASSSM, volume 2)
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About this book
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.
Keywords
Table of contents (7 chapters)
Authors and Affiliations
About the author
He previously held positions at University of Wisconsin-Madison. He received Ho-Am Prize in Science in 2022.
Bibliographic Information
Book Title: Lagrangian Floer Theory and Its Deformations
Book Subtitle: An Introduction to Filtered Fukaya Category
Authors: Yong-Geun Oh
Series Title: KIAS Springer Series in Mathematics
DOI: https://doi.org/10.1007/978-981-97-1798-9
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
Hardcover ISBN: 978-981-97-1797-2Published: 07 June 2024
Softcover ISBN: 978-981-97-1800-9Due: 21 June 2025
eBook ISBN: 978-981-97-1798-9Published: 06 June 2024
Series ISSN: 2731-5142
Series E-ISSN: 2731-5150
Edition Number: 1
Number of Pages: XVII, 416
Number of Illustrations: 29 b/w illustrations, 1 illustrations in colour
Topics: Differential Geometry, Algebraic Topology, Topology