Symplectic Geometry and Its Applications
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Gromov Receives 2009 Abel Prize
Gromov Receives 2009 Abel Prize . The Norwegian Academy of Science Medal (1997), and the Wolf Prize (1993). He is a and Letters has decided to award the foreign member of the U.S. National Academy of Abel Prize for 2009 to the Russian- Sciences and of the American Academy of Arts French mathematician Mikhail L. and Sciences, and a member of the Académie des Gromov for “his revolutionary con- Sciences of France. tributions to geometry”. The Abel Prize recognizes contributions of Citation http://www.abelprisen.no/en/ extraordinary depth and influence Geometry is one of the oldest fields of mathemat- to the mathematical sciences and ics; it has engaged the attention of great mathema- has been awarded annually since ticians through the centuries but has undergone Photo from from Photo 2003. It carries a cash award of revolutionary change during the last fifty years. Mikhail L. Gromov 6,000,000 Norwegian kroner (ap- Mikhail Gromov has led some of the most impor- proximately US$950,000). Gromov tant developments, producing profoundly original will receive the Abel Prize from His Majesty King general ideas, which have resulted in new perspec- Harald at an award ceremony in Oslo, Norway, on tives on geometry and other areas of mathematics. May 19, 2009. Riemannian geometry developed from the study Biographical Sketch of curved surfaces and their higher-dimensional analogues and has found applications, for in- Mikhail Leonidovich Gromov was born on Decem- stance, in the theory of general relativity. Gromov ber 23, 1943, in Boksitogorsk, USSR. He obtained played a decisive role in the creation of modern his master’s degree (1965) and his doctorate (1969) global Riemannian geometry. -
Covariant Hamiltonian Field Theory 3
December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte COVARIANT HAMILTONIAN FIELD THEORY JURGEN¨ STRUCKMEIER and ANDREAS REDELBACH GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstr. 1, 64291 Darmstadt, Germany and Johann Wolfgang Goethe-Universit¨at Frankfurt am Main Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany [email protected] Received 18 July 2007 Revised 14 December 2020 A consistent, local coordinate formulation of covariant Hamiltonian field theory is pre- sented. Whereas the covariant canonical field equations are equivalent to the Euler- Lagrange field equations, the covariant canonical transformation theory offers more gen- eral means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinites- imal canonical transformation furnishes the most general form of Noether’s theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations. Keywords: Field theory; Hamiltonian density; covariant. PACS numbers: 11.10.Ef, 11.15Kc arXiv:0811.0508v6 [math-ph] 15 Dec 2020 1. Introduction Relativistic field theories and gauge theories are commonly formulated on the basis of a Lagrangian density L1,2,3,4. The space-time evolution of the fields is obtained by integrating the Euler-Lagrange field equations that follow from the four-dimensional representation of Hamilton’s action principle. -
“Generalized Complex and Holomorphic Poisson Geometry”
“Generalized complex and holomorphic Poisson geometry” Marco Gualtieri (University of Toronto), Ruxandra Moraru (University of Waterloo), Nigel Hitchin (Oxford University), Jacques Hurtubise (McGill University), Henrique Bursztyn (IMPA), Gil Cavalcanti (Utrecht University) Sunday, 11-04-2010 to Friday, 16-04-2010 1 Overview of the Field Generalized complex geometry is a relatively new subject in differential geometry, originating in 2001 with the work of Hitchin on geometries defined by differential forms of mixed degree. It has the particularly inter- esting feature that it interpolates between two very classical areas in geometry: complex algebraic geometry on the one hand, and symplectic geometry on the other hand. As such, it has bearing on some of the most intriguing geometrical problems of the last few decades, namely the suggestion by physicists that a duality of quantum field theories leads to a ”mirror symmetry” between complex and symplectic geometry. Examples of generalized complex manifolds include complex and symplectic manifolds; these are at op- posite extremes of the spectrum of possibilities. Because of this fact, there are many connections between the subject and existing work on complex and symplectic geometry. More intriguing is the fact that complex and symplectic methods often apply, with subtle modifications, to the study of the intermediate cases. Un- like symplectic or complex geometry, the local behaviour of a generalized complex manifold is not uniform. Indeed, its local structure is characterized by a Poisson bracket, whose rank at any given point characterizes the local geometry. For this reason, the study of Poisson structures is central to the understanding of gen- eralized complex manifolds which are neither complex nor symplectic. -
Hamiltonian and Symplectic Symmetries: an Introduction
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 3, July 2017, Pages 383–436 http://dx.doi.org/10.1090/bull/1572 Article electronically published on March 6, 2017 HAMILTONIAN AND SYMPLECTIC SYMMETRIES: AN INTRODUCTION ALVARO´ PELAYO In memory of Professor J.J. Duistermaat (1942–2010) Abstract. Classical mechanical systems are modeled by a symplectic mani- fold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called sym- plectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton’s equations. Since then a number of connections with combinatorics, finite- dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent re- sults on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry. 1. Introduction Symplectic geometry is concerned with the study of a notion of signed area, rather than length, distance, or volume. It can be, as we will see, less intuitive than Euclidean or metric geometry and it is taking mathematicians many years to understand its intricacies (which is work in progress). The word “symplectic” goes back to the 1946 book [164] by Hermann Weyl (1885–1955) on classical groups. -
Moon-Earth-Sun: the Oldest Three-Body Problem
Moon-Earth-Sun: The oldest three-body problem Martin C. Gutzwiller IBM Research Center, Yorktown Heights, New York 10598 The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler’s laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today’s many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. -
Symplectic Geometry
Part III | Symplectic Geometry Based on lectures by A. R. Pires Notes taken by Dexter Chua Lent 2018 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. The first part of the course will be an overview of the basic structures of symplectic ge- ometry, including symplectic linear algebra, symplectic manifolds, symplectomorphisms, Darboux theorem, cotangent bundles, Lagrangian submanifolds, and Hamiltonian sys- tems. The course will then go further into two topics. The first one is moment maps and toric symplectic manifolds, and the second one is capacities and symplectic embedding problems. Pre-requisites Some familiarity with basic notions from Differential Geometry and Algebraic Topology will be assumed. The material covered in the respective Michaelmas Term courses would be more than enough background. 1 Contents III Symplectic Geometry Contents 1 Symplectic manifolds 3 1.1 Symplectic linear algebra . .3 1.2 Symplectic manifolds . .4 1.3 Symplectomorphisms and Lagrangians . .8 1.4 Periodic points of symplectomorphisms . 11 1.5 Lagrangian submanifolds and fixed points . 13 2 Complex structures 16 2.1 Almost complex structures . 16 2.2 Dolbeault theory . 18 2.3 K¨ahlermanifolds . 21 2.4 Hodge theory . 24 3 Hamiltonian vector fields 30 3.1 Hamiltonian vector fields . 30 3.2 Integrable systems . 32 3.3 Classical mechanics . 34 3.4 Hamiltonian actions . 36 3.5 Symplectic reduction . 39 3.6 The convexity theorem . 45 3.7 Toric manifolds . 51 4 Symplectic embeddings 56 Index 57 2 1 Symplectic manifolds III Symplectic Geometry 1 Symplectic manifolds 1.1 Symplectic linear algebra In symplectic geometry, we study symplectic manifolds. -
Symplectic Geometry Tara S
THE GRADUATE STUDENT SECTION WHAT IS... Symplectic Geometry Tara S. Holm Communicated by Cesar E. Silva In Euclidean geome- depending smoothly on the point 푝 ∈ 푀.A 2-form try in a vector space 휔 ∈ Ω2(푀) is symplectic if it is both closed (its exterior Symplectic over ℝ, lengths and derivative satisfies 푑휔 = 0) and nondegenerate (each structures are angles are the funda- function 휔푝 is nondegenerate). Nondegeneracy is equiva- mental measurements, lent to the statement that for each nonzero tangent vector floppier than and objects are rigid. 푣 ∈ 푇푝푀, there is a symplectic buddy: a vector 푤 ∈ 푇푝푀 In symplectic geome- so that 휔푝(푣, 푤) = 1.A symplectic manifold is a (real) holomorphic try, a two-dimensional manifold 푀 equipped with a symplectic form 휔. area measurement is Nondegeneracy has important consequences. Purely in functions or the key ingredient, and terms of linear algebra, at any point 푝 ∈ 푀 we may choose the complex numbers a basis of 푇푝푀 that is compatible with 휔푝, using a skew- metrics. are the natural scalars. symmetric analogue of the Gram-Schmidt procedure. We It turns out that sym- start by choosing any nonzero vector 푣1 and then finding a plectic structures are much floppier than holomorphic symplectic buddy 푤1. These must be linearly independent functions in complex geometry or metrics in Riemannian by skew-symmetry. We then peel off the two-dimensional geometry. subspace that 푣1 and 푤1 span and continue recursively, The word “symplectic” is a calque introduced by eventually arriving at a basis Hermann Weyl in his textbook on the classical groups. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
Some Aspects of the Geodesic Flow
Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael’s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques from Symplectic Geometry. 1 Introduction We begin defining our object of study. Definition 1.1. Given a closed manifold M with Riemannian metric g, the co- geodesic flow is defined as the Hamiltonian flow φt on the cotangent bundle (T ∗M,Θ)(Θ will always denote the canonical form on T ∗M) where the cor- 1 responding lagrangian is L = 2 g Suppose that the Legendre transform is given by (x; v) 7! (x; p). Then we know • H = hp; vi − L(x; v) • p is given as the fiber derivative of L. Namely, consider f(v) = L(x; v) for x fixed and then p(x,v) is given by p = Dvf 1 In our very particular case, we can easily calculate the transform using Eüler’s lemma for homogeneous functions (L is just a quadratic form here); ge wet hp; vi = hDvf ; vi = 2f(v) = 2L = g(v; v) This implies • the Legendre transform identifies the fibers of TM and T ∗M using the metric (v 7! g(v; ·)). • H = 2L − L = L. This shows that the co-geodesic flow induces a flow (which we will continue denoting φt) on TM such that φt = (γ; γ_ ) and γ satisfies the variational equation for the action defined by L = H. 1 d @L We have that dt @v = 0, so the norm of γ_ is constant. -
7.5 Lagrange Brackets
Chapter 7 Brackets Module 1 Poisson brackets and Lagrange brackets 1 Brackets are a powerful and sophisticated tool in Classical mechanics particularly in Hamiltonian formalism. It is a way of characterization of canonical transformations by using an operation, known as bracket. It is very useful in mathematical formulation of classical mechanics. It is a powerful geometrical structure on phase space. Basically it is a mapping which maps two functions into a new function. Reformulation of classical mechanics is indeed possible in terms of brackets. It provides a direct link between classical and quantum mechanics. 7.1 Poisson brackets Poisson bracket is an operation which takes two functions of phase space and time and produces a new function. Let us consider two functions F1(,,) qii p t and F2 (,,) qii p t . The Poisson bracket is defined as n FFFF1 2 1 2 FF12,. (7.1) i1 qi p i p i q i For single degree of freedom, Poisson bracket takes the following form FFFF FF,. 1 2 1 2 12 q p p q Basically in Poisson bracket, each term contains one derivative of F1 and one derivative of F2. One derivative is considered with respect to a coordinate q and the other is with respect to the conjugate momentum p. A change of sign occurs in the terms depending on which function is differentiated with respect to the coordinate and which with respect to the momentum. Remark: Some authors use square brackets to represent the Poisson bracket i.e. instead ofFF12, , they prefer to use FF12, . Example 1: Let F12( q , p ) p q , F ( q , p ) sin q . -
Symplectic Geometry
SYMPLECTIC GEOMETRY Gert Heckman Radboud University Nijmegen [email protected] Dedicated to the memory of my teacher Hans Duistermaat (1942-2010) 1 Contents Preface 3 1 Symplectic Linear Algebra 5 1.1 Symplectic Vector Spaces . 5 1.2 HermitianForms ......................... 7 1.3 ExteriorAlgebra ......................... 8 1.4 TheWord“Symplectic” . 10 1.5 Exercises ............................. 11 2 Calculus on Manifolds 14 2.1 Vector Fields and Flows . 14 2.2 LieDerivatives .......................... 15 2.3 SingularHomology . .. .. .. .. .. .. 18 2.4 Integration over Singular Chains and Stokes Theorem . 20 2.5 DeRhamTheorem........................ 21 2.6 Integration on Oriented Manifolds and Poincar´eDuality . 22 2.7 MoserTheorem.......................... 25 2.8 Exercises ............................. 27 3 Symplectic Manifolds 29 3.1 RiemannianManifolds . 29 3.2 SymplecticManifolds. 29 3.3 FiberBundles........................... 32 3.4 CotangentBundles . .. .. .. .. .. .. 34 3.5 GeodesicFlow .......................... 37 3.6 K¨ahlerManifolds . 40 3.7 DarbouxTheorem ........................ 43 3.8 Exercises ............................. 46 4 Hamilton Formalism 49 4.1 PoissonBrackets ......................... 49 4.2 IntegrableSystems . 51 4.3 SphericalPendulum . .. .. .. .. .. .. 55 4.4 KeplerProblem.......................... 61 4.5 ThreeBodyProblem. .. .. .. .. .. .. 65 4.6 Exercises ............................. 66 2 5 Moment Map 69 5.1 LieGroups ............................ 69 5.2 MomentMap ........................... 74 5.3 SymplecticReduction . 80 5.4 Symplectic Reduction for Cotangent Bundles . 84 5.5 Geometric Invariant Theory . 89 5.6 Exercises ............................. 93 3 Preface These are lecture notes for a course on symplectic geometry in the Dutch Mastermath program. There are several books on symplectic geometry, but I still took the trouble of writing up lecture notes. The reason is that this one semester course was aiming for students at the beginning of their masters. -
A Symplectic Fixed Point Theorem on Open Manifolds
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 4, April 1982 A SYMPLECTIC FIXED POINT THEOREM ON OPEN MANIFOLDS MICHAEL COLVIN AND KENT MORRISON Abstract. In 1968 Bourgin proved that every measure-preserving, orientation- preserving homeomorphism of the open disk has a fixed point, and he asked whether such a result held in higher dimensions. Asimov, in 1976, constructed counterexam- ples in all higher dimensions. In this paper we answer a weakened form of Bourgin's question dealing with symplectic diffeomorphisms: every symplectic diffeomorphism of an even-dimensional cell sufficiently close to the identity in the C'-fine topology has a fixed point. This result follows from a more general result on open manifolds and symplectic diffeomorphisms. Introduction. Fixed point theorems for area-preserving mappings have a history which dates back to Poincaré's "last geometric theorem", i.e., any area-preserving mapping of an annulus which twists the boundary curves in opposite directions has at least two fixed points. More recently it has been proved that any area-preserving, orientation-preserving mapping of the two-dimensional sphere into itself possesses at least two distinct fixed points (see [N, Si]). In the setting of noncompact manifolds, Bourgin [B] showed that any measure-preserving, orientation-preserving homeomor- phism of the open two-cell B2 has a fixed point. For Bourgin's theorem one assumes that the measure is finite on B2 and that the measure of a nonempty open set is positive. Bourgin also gave a counterexample to the generalization of the theorem for the open ball in R135and asked the question whether his theorem remains valid for the open balls in low dimensions.