Symplectic topology notes pdf

Lectures on symplectic geometry ana cannas da silva1 revised january 2006 published by springerverlag as number 1764 of the series lecture notes in mathematics. Symplectic topology and floer homology by yonggeun oh. This site is like a library, use search box in the widget to get ebook that you want. Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact manifolds. Then we introduce the fukaya category informally and without a lot of the necessary technical detail, and briefly discuss algebraic concepts such as exact triangles and. Having been latexd, they may look good, but in fact they are in no way. In addition to the notes by cannas da silva mentioned by dick palais, here are further two advanced books covering somewhat different territory. Symplectic topology and capacities1 stony brook mathematics. The book was the first comprehensive introduction to the subject and became a key text in the area.

The study of symplectic manifolds is called symplectic geometry or symplectic topology. Twenty years ago symplectic topology did not exist. Since these forms are closed, they are locally exact, and hence up to some coverings of xand g, we may assume these form are exact. Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as springs, planetary motion, and wave propagation. Lectures on symplectic geometry ana cannas da silva springer. Over the last few years powerful new methods in analysis and topology have led to the development of the modern global theory ofsymplectic topology, including several striking and important results.

Differential forms gi ven ma smooth manifold, on has two natural bundles. The tangent space tpm is the linearization of the nonlinear space m at the point p p m. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. A significantly revised second edition was published in 1998 introducing new sections and updates on the fastdeveloping area.

Some parts of these notes are taken from the survey article 31. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures. From dynamics to contact and symplectic topology and back. Approximately holomorphic methods in symplectic topology denisauroux lecturenotesandbibliography 1. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost. Faculty of mathematics and computer science, university of science,vietnamnationaluniversity,227nguyenvancu,district5,hochiminh city, vietnam. Pires notes taken by dexter chua lent 2018 these notes are not endorsed by the lecturers, and i have modi ed them often signi cantly after lectures.

Introduction asymplecticstructureonasmoothmanifoldisaclosednon. Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from floer theory as a whole. Notes on smooth topology and symplectic embedding problems these notes were written for participants in the symplectic geometry group in the 2018 berkeley geometry reu. For a time, people wondered whether every symplectic manifold was k ahler. Elementary symplectic topology and mechanics franco cardin. Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the relations between the quantities entering into the theory. Symplectic geometry is the geometry of symplectic manifolds. These are lecture notes for a course on symplectic geometry in the dutch. Click download or read online button to get lectures on symplectic manifolds book now.

A symplectic manifold is a 2nmanifold m2n with an atlas of charts such that the derivatives of transition maps are in sp 2n r. Lectures on symplectic geometry lecture notes in mathematics. Weve classied topological fourmanifolds, but not smooth ones, so symplectic topology is a very useful tool for this. The first edition of introduction to symplectic topology was published in 1995. A symplectic form on v is a skewsymmetric bilinear nondegen. Symplectic topology today joint mathematics meetings. Dusa mcduff and dietmar salamon, introduction to symplectic topology. Lectures on categorical dynamics and symplectic topology. Lecture notes on symplectic geometry symplectic geometry. Michele audin, torus actions on symplectic manifolds 2nd edition a. The goal of these notes is to give a short introduction to fukaya categories and some of their applications. Introduction to symplectic topology dusa mcduff, dietmar.

The symplectic topology of ramanujams surface article pdf available in commentarii mathematici helvetici 804 december 2004 with 25 reads how we measure reads. Symplectic geometry is the mathematical apparatus of such areas of physics as classical mechanics, geometrical optics and thermodynamics. The tangent bundle tm n m of a smooth manifold m is the. In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, m \displaystyle m, equipped with a closed nondegenerate differential 2form. An introduction to symplectic topology through sheaf theory c. Robert gompf ut, 19934 showed that there are many examples of nonk ahler symplectic manifolds. Thus symplectic geometry is essentially topological in nature. Lectures on symplectic manifolds download ebook pdf, epub. Symplectic topology example sheet 1 dietmar salamon eth zuric h 27 february 20 exercise 1. 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. Symplectic linear algebra let v be a real vector space. Part iii symplectic geometry based on lectures by a. The paper begins with symplectic manifolds and their lagrangian submanifolds, covers contact manifolds and their legendrian submanifolds, and indicates the first steps of symplectic and contact topology. An introduction to symplectic geometry megumi harada, lecture notes by kaname hashimoto summer school, osaka, 2007 abstract the intent of this series of lectures is twofold.

Symplectic geometry lecture notes, university of toronto. Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations. Symplectic structures underlie the equations of classical mechanics and their properties are reflected in the behaviour of a wide range of physical systems. This book is a slightly expanded version of the lecture notes i produced for a twosemester course taught at university college london in 201516, for ph. The goal of these notes is to provide a fast introduction to symplectic geometry. An introduction to symplectic topology through sheaf theory. Lecture notes of symplectic topology and floer homology. As a quick reference for young researchers in mathematics who already master the rudiments of symplectic topology and geometry it might be an inexpensive possibility. A symplectic form on v is a skewsymmetric bilinear nondegen erate form. I thank hao yin for allowing me to use his lecture notes from beijing. Write the elements of the con guration space r2 n rn r in the form z x.

Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the. Our task is made a little easier by the fact that symplectic topology was only born around 1983. Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes. Introduction to symplectic topology oxford graduate texts in. Published by springerverlag as number 1764 of the series lecture notes in.

Symplectic topology and algebraic geometry interact in many fruitful ways. It is now understood to arise naturally in algebraic geometry, in lowdimensional topology, in representation theory and in string theory. Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the relations between the quantities. Note that gromov can only establish uniqueness under the assumption that the manifold z contains a symplectic 2sphere s. Notes on symplectic geometry 5 conversely, suppose that x is a symplectic manifold with a transitive. These are the notes for an advanced graduate course given at mit in spring 20. The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of hamiltonian systems. Symplectic topology and geometric quantum mechanics.

As a textbook i cannot recommend these lectures on symplectic geometry, as they are too dense, too prooforiented and too exampledevoid. The aim of this course is to present the recent work connecting sheaf theory and symplectic topology, due to several authors, nadler nad, nadz. Lectures on symplectic geometry ana cannas da silva. Our main sources have been the books symplectic techniques by guilleminsternberg and introduction to symplectic topology by mcduffsalamon, and the paper stratified symplectic spaces and reduction, ann. A symplectic manifold is a manifold equipped with a symplectic form.