An introduction to symplectic geometry 3 we now turn to manifolds. Problem sheets are being prepared by terry farrelly and will be made available here on my blog. R is a skewsymmetric bilinear map and w p varies smoothly in p. This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at eth zurich and the humboldt university berlin in 20092010. There are several books on symplectic geometry, but i still took the trouble of writing up lecture notes. Symplectic forms allow the denition of symplectic bases, which are the analogues of orthonormal bases in euclidean geometry.
Lectures on symplectic geometry fraydoun rezakhanlou departmet of mathematics, uc berkeley september 18, 2018 chapter 1. For an upperlevel undergraduate or beginning graduate student, lectures on symplectic geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics. An introduction to symplectic topology through sheaf theory. Lectures on symplectic geometry ana cannas da silva1 revised january 2006 published by springerverlag as number 1764 of the series lecture notes in mathematics. It turns out that this geometric approach to hamiltonian systems is very fruitful. In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, equipped with a closed nondegenerate differential 2form, called the symplectic form. The central piece of structure in symplectic geometry is a closed. Quantitative symplectic geometry 3 symplectic geometry is the geometry underlying hamiltonian systems. For a more liegroup focused account, you can try robert bryants lectures on lie groups and symplectic geometry which are available online here. Symplectic geometry nicholas proudfoot department of mathematics, university of oregon, eugene, or 97403 these notes are written for a ten week graduate class on symplectic geometry. In the final lecture he describes the hprinciple and others ideas of gromov in symplectic geometry, like pseudoholomorphic curves.
Salamon, introduction to symplectic topology, oxford university press, new york, 1998. Lectures on symplectic geometry pdf 225p this note contains on the following subtopics of symplectic geometry, symplectic manifolds, symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, kahler manifolds, hamiltonian mechanics, moment maps, symplectic reduction, moment maps revisited and symplectic toric manifolds. Finally, two basic references on quantum mechanics itself are and 20. For winter semester 201718 i am giving a course on symplectic geometry and classical mechanics. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and. Lectures on symplectic geometry fraydoun rezakhanlou departmet of mathematics, uc berkeley december 12, 2015 chapter 1. This volume contains a collection of papers based on lectures delivered by distinguished mathematicians at clay mathematics institute events over the past few years. This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for phd students given in 201516. Anintroductiontoliegroups andsymplecticgeometry a series of nine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in park city, utah, 24 june20 july 1991. Recall from last time the statement of the following lemma.
This course is intended for anyone with a familiarity with c. This is an other great mathematics book cover the following topics of problem solving. Although not explicitly linked, the topics in this inaugural volume have a common flavour and a common appeal to all who are interested in recent. An introduction to symplectic geometry springerlink.
Symplectic geometry is a central topic of current research in mathematics. Symplectic geometry of rationally connected threefolds tian, zhiyu, duke mathematical journal, 2012. Explicit examples are discussed in section 2 below. Symplectic and contact geometry summer tutorial 2003. The updated version can be downloaded from the authors website. To see an extisive list of symplectic geometry ebooks. Differential forms gi ven ma smooth manifold, on has two natural bundles. In this section we want to recall some standard definitions and properties of symplectic geometry. The notes are still incomplete, but due to recent requests from readers, ive decided to make a presentable halffinished version available here. This means that the technical groundwork is largely of the general categorical kind, more precisely taking place in the framework of a 1categories. Denis auroux recall from last time the statement of the following lemma. Symplectic manifolds and darbouxs theorem chapter 4.
The subject of the lectures is symmetries of fukaya categories. Lecture course introduction to symplectic geometry janko latschev winter semester 201011 according to kleins view as expressed in his famous erlangen program, geometry is the study of certain structures on spaces, together with their transformation groups. Motivation is drawn from algebraic geometry and mirror symmetry, but the ultimate interest is in applications to symplectic topology. Symplectic forms skewsymmetric bilinear maps, symplectic vector spaces, symplectic manifolds, symplectomorphisms, homework 1. Symplectic geometry has its origins in the hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of lie groups. It is intended to be the first in an occasional series of volumes of cmi lectures. An introduction to symplectic geometry alessandro assef institute for theoretical physics university of cologne these notes are a short sum up about two talks that i gave in august and september 2015 an the university of cologne in my workgroup seminar by prof. Introduction to symplectic geometry solution for the midterm thursday, november 7, 20 1truefalse questions, no justications needed. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, kaehler manifolds, hamiltonian mechanics. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill in gaps in the existing literature where necessary, and then gives detailed explanations of a few of the standard applications in contact.
This course is intended for anyone with a familiarity with classical mechanics and basic. These are lecture notes for a course on symplectic geometry in the dutch mastermath program. Cannas da silva, lectures on symplectic geometry, springer verlag, berlin, heidelberg, 2001. We will also use material from some additional sources, such as. We hope mathematician or person whos interested in mathematics like these books. Lectures on categorical dynamics and symplectic topology. This are the notes of graduate lectures given in the fall semester 2010 at princeton university, and then as the eilenberg lectures at columbia in the spring 2011.
The presence of symmetries in a classical system translates in the context of symplectic geometry in the presence of an action of a lie group g on the phase space m. Most of the material here is included in mich ele audins book torus actions on symplectic manifolds, which i used heavily in preparing these notes. Lectures on symplectic geometry written by ana cannas da silva. This course is intended for anyone with a familiarity with classical mechanics and basic differential geometry. 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. Contact manifolds and weinstein conjecture chapter 5. The study of symplectic manifolds is called symplectic geometry or symplectic topology. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography.
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