Morse Theory and Floer Homology

Author: Michèle Audin
Publisher: Springer Science & Business Media
ISBN: 9781447154969
Release Date: 2013-11-29
Genre: Mathematics

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.

Differentialgeometrie und Minimalfl chen

Author: Jürgen Jost
Publisher: Springer-Verlag
ISBN: 9783662067185
Release Date: 2013-03-09
Genre: Mathematics

Das vorliegende Lehrbuch bietet eine moderne Einführung in die Differentialgeometrie etwa im Umfang einer einsemestrigen Vorlesung. Zunächst wird die Geometrie von Flächen im Raum behandelt. Hierbei wird die geometrische Anschauung des Lesers anhand vieler Beispiele gefördert, deren wichtigste Klasse die Minimalflächen bilden. Zu ihrem Studium werden analytische Methoden entwickelt, und in diesem Zusammenhang wird auch das Plateausche Problem, eine Minimalfläche mit vorgegebener Berandung zu finden, gelöst. Als Beispiel einer globalen Aussage der Differentialgeometrie wird der Bernsteinsche Satz bewiesen. Weitere Kapitel behandeln die innere Geometrie von Flächen, einschließlich des Satzes von Gauss-Bonnet und einer ausführlichen Darstellung der hyperbolischen Geometrie. Verschiedene geistesgeschichtliche Bemerkungen runden diesen Text ab, welcher durch seine Verbindung von geometrischen Konstruktionen und analytischen Methoden einem zentralen Trend der modernen mathematischen Forschung folgt. Das erste Lehrbuch, das eine gründliche Einführung in die Theorie der Minimalflächen gewährleistet.

Geometrical Themes Inspired by the N body Problem

Author: Luis Hernández-Lamoneda
Publisher: Springer
ISBN: 9783319714288
Release Date: 2018-02-26
Genre: Mathematics

Presenting a selection of recent developments in geometrical problems inspired by the N-body problem, these lecture notes offer a variety of approaches to study them, ranging from variational to dynamical, while developing new insights, making geometrical and topological detours, and providing historical references. A. Guillot’s notes aim to describe differential equations in the complex domain, motivated by the evolution of N particles moving on the plane subject to the influence of a magnetic field. Guillot studies such differential equations using different geometric structures on complex curves (in the sense of W. Thurston) in order to find isochronicity conditions. R. Montgomery’s notes deal with a version of the planar Newtonian three-body equation. Namely, he investigates the problem of whether every free homotopy class is realized by a periodic geodesic. The solution involves geometry, dynamical systems, and the McGehee blow-up. A novelty of the approach is the use of energy-balance in order to motivate the McGehee transformation. A. Pedroza’s notes provide a brief introduction to Lagrangian Floer homology and its relation to the solution of the Arnol’d conjecture on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism.

Perturbed Gradient Flow Trees and A algebra Structures in Morse Cohomology

Author: Stephan Mescher
Publisher: Springer
ISBN: 9783319765846
Release Date: 2018-04-25
Genre: Mathematics

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.

Lectures on Morse Homology

Author: Augustin Banyaga
Publisher: Springer Science & Business Media
ISBN: 1402026951
Release Date: 2004-10-29
Genre: Mathematics

This book offers a detailed presentation of results needed to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. The text presents results that were formerly scattered in the mathematical literature, in a single reference with complete and detailed proofs. The core material includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory.

Symplectic Topology and Floer Homology

Author: Yong-Geun Oh
Publisher: Cambridge University Press
ISBN: 9781107072459
Release Date: 2015-08-27
Genre: Mathematics

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.

Elementary Symplectic Topology and Mechanics

Author: Franco Cardin
Publisher: Springer
ISBN: 9783319110264
Release Date: 2014-12-01
Genre: Science

This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.

Partielle Differentialgleichungen der Geometrie und der Physik 2

Author: Friedrich Sauvigny
Publisher: Springer-Verlag
ISBN: 9783540275404
Release Date: 2005-12-05
Genre: Mathematics

Das zweibändige Lehrbuch behandelt das Gebiet der partiellen Differentialgleichungen umfassend und anschaulich. Der Autor stellt in Band 2 funktionalanalytische Lösungsmethoden vor und erläutert u. a. die Lösbarkeit von Operatorgleichungen im Banachraum, lineare Operatoren im Hilbertraum und Spektraltheorie, die Schaudersche Theorie linearer elliptischer Differentialgleichungen sowie schwache Lösungen elliptischer Differentialgleichungen.

Riemannian Geometry and Geometric Analysis

Author: Jürgen Jost
Publisher: Springer Science & Business Media
ISBN: 9783642212987
Release Date: 2011-07-28
Genre: Mathematics

This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry. The 6th edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the book. From the reviews: "This book provides a very readable introduction to Riemannian geometry and geometric analysis. ... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome." Mathematical Reviews "...the material ... is self-contained. Each chapter ends with a set of exercises. Most of the paragraphs have a section ‘Perspectives’, written with the aim to place the material in a broader context and explain further results and directions." Zentralblatt MATH

Vorlesungen ber die Zahlentheorie der Quaternionen

Author: Adolf Hurwitz
Publisher: Springer-Verlag
ISBN: 9783642475368
Release Date: 2013-03-13
Genre: Mathematics

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Gew hnliche Differentialgleichungen

Author: Vladimir I. Arnold
Publisher: Springer-Verlag
ISBN: 9783642564802
Release Date: 2013-03-11
Genre: Mathematics

nen (die fast unverändert in moderne Lehrbücher der Analysis übernommen wurde) ermöglichten ihm nach seinen eigenen Worten, "in einer halben Vier telstunde" die Flächen beliebiger Figuren zu vergleichen. Newton zeigte, daß die Koeffizienten seiner Reihen proportional zu den sukzessiven Ableitungen der Funktion sind, doch ging er darauf nicht weiter ein, da er zu Recht meinte, daß die Rechnungen in der Analysis bequemer auszuführen sind, wenn man nicht mit höheren Ableitungen arbeitet, sondern die ersten Glieder der Reihenentwicklung ausrechnet. Für Newton diente der Zusammenhang zwischen den Koeffizienten der Reihe und den Ableitungen eher dazu, die Ableitungen zu berechnen als die Reihe aufzustellen. Eine von Newtons wichtigsten Leistungen war seine Theorie des Sonnensy stems, die in den "Mathematischen Prinzipien der Naturlehre" ("Principia") ohne Verwendung der mathematischen Analysis dargestellt ist. Allgemein wird angenommen, daß Newton das allgemeine Gravitationsgesetz mit Hilfe seiner Analysis entdeckt habe. Tatsächlich hat Newton (1680) lediglich be wiesen, daß die Bahnkurven in einem Anziehungsfeld Ellipsen sind, wenn die Anziehungskraft invers proportional zum Abstandsquadrat ist: Auf das Ge setz selbst wurde Newton von Hooke (1635-1703) hingewiesen (vgl. § 8) und es scheint, daß es noch von weiteren Forschern vermutet wurde.

Was immer du willst

Author: Shelby Reed
Publisher: Heyne Verlag
ISBN: 9783641167912
Release Date: 2015-10-12
Genre: Fiction

Billie Cort soll für die Zeitschrift Illicit einen Artikel über den exklusiven Male-Escort-Club Avalon schreiben. Bis zu 1000 Dollar bezahlen hier Frauen für eine Nacht mit dem Mann ihrer Wahl. Sogar Adrian, der begehrteste Mann im Illicit, wird ihr für ein kurzes Interview zur Verfügung stehen. Die Begegnung mit ihm verändert das Leben der schüchternen Reporterin. Denn noch nie hat die sie ein solches Verlangen gespürt wie für den verschlossenen Escort mit den dunklen Augen ...

Dirichlet Forms and Analysis on Wiener Space

Author: Nicolas Bouleau
Publisher: Walter de Gruyter
ISBN: 9783110858389
Release Date: 1991-01-01
Genre: Mathematics

The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima’s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss “carré du champ” operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)