Topology of Symplectomorphism Groups and Ball-Swappings
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Density of Thin Film Billiard Reflection Pseudogroup in Hamiltonian Symplectomorphism Pseudogroup Alexey Glutsyuk
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk To cite this version: Alexey Glutsyuk. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomor- phism pseudogroup. 2020. hal-03026432v2 HAL Id: hal-03026432 https://hal.archives-ouvertes.fr/hal-03026432v2 Preprint submitted on 6 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk∗yzx December 3, 2020 Abstract Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C1-smooth hypersurface γ ⊂ Rn+1 that is either a global strictly convex closed hypersurface, or a germ of hy- persurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deforma- tions. In the case, when γ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case, when γ is a germ. -
Lecture 1: Basic Concepts, Problems, and Examples
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic geometry/topology, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. -
Fall 2017 MATH 70330 “Intermediate Geometry and Topology” Pavel Mnev Detailed Plan. I. Characteristic Classes. (A) Fiber/Vec
Fall 2017 MATH 70330 \Intermediate Geometry and Topology" Pavel Mnev Detailed plan. I. Characteristic classes. (a) Fiber/vector/principal bundles. Examples. Connections, curvature. Rie- mannian case. Ref: [MT97]. (b) (2 classes.) Stiefel-Whitney, Chern and Pontryagin classes { properties, axiomatic definition. Classifying map and classifying bundle. RP 1; CP 1, infinite Grassmanians Gr(n; 1), CW structure, cohomology ring (univer- sal characteristic classes). Characteristic classes as obstructions (e.g. for embeddability of projective spaces). Euler class. Also: Chern character, splitting principle, Chern roots. Also: BG for a finite group/Lie group, group cohomology. Ref: [MS74]; also: [BT82, Hat98, LM98, May99]. (c) Chern-Weil homomorphism. Example: Chern-Gauss-Bonnet formula. Chern- Simons forms. Ref: Appendix C in [MS74]; [AM05, Dup78, MT97]. (d) Equivariant cohomology. Borel model (homotopy quotient), algebraic ver- sion { Cartan and Weil models. Ref: [GS99, Mei06, Tu13]. II. Bits of symplectic geometry. (a) Symplectic linear algebra, Lagrangian Grassmanian, Maslov class. Ref: [BW97, Ran, MS17]. (b) Symplectic manifolds, Darboux theorem (proof via Moser's trick). Dis- tinguished submanifolds (isotropic, coisotropic, Lagrangian). Examples. Constructions of Lagrangians in T ∗M (graph, conormal bundle). Ref: [DS00]. (c) Hamiltonian group actions, moment maps, symplectic reduction. Ref: [Jef] (lectures 2{4,7), [DS00]. (d) (Optional) Convexity theorem (Atyiah-Guillemin-Sternberg) for the mo- ment map of a Hamiltonian torus action. Toric varieties as symplectic re- ductions, their moment polytopes, Delzant's theorem. Ref: [Pra99, Sch], Part XI in [DS00], lectures 5,6 in [Jef]. (e) Localization theorems: Duistermaat-Heckman and Atiyah-Bott. Ref: [Mei06, Tu13], [Jef] (lecture 10). (f) (Optional) Classical field theory via Lagrangian correspondences. -
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020 Lectures by R. Inanç˙ Baykur University of Massachusetts Amherst Disclaimer These notes were taken during lecture using the vimtex package of the editor neovim. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 January 21 • 5 1.1 Course Description • 5 1.2 Organization • 5 1.2.1 Notational conventions•5 1.3 Basic Notions • 5 1.4 Symplectic Linear Algebra • 6 2 January 23 • 8 2.1 More Basic Linear Algebra • 8 2.2 Compatible Complex Structures and Inner Products • 9 3 January 28 • 10 3.1 A Big Theorem • 10 3.2 More Compatibility • 11 4 January 30 • 13 4.1 Homework Exercises • 13 4.2 Subspaces of Symplectic Vector Spaces • 14 5 February 4 • 16 5.1 Linear Algebra, Conclusion • 16 5.2 Symplectic Vector Bundles • 16 6 February 6 • 18 6.1 Proof of Theorem 5.11 • 18 6.2 Vector Bundles, Continued • 18 6.3 Compatible Triples on Manifolds • 19 7 February 11 • 21 7.1 Obtaining Compatible Triples • 21 7.2 Complex Structures • 21 8 February 13 • 23 2 3 8.1 Kähler Forms Continued • 23 8.2 Some Algebraic Geometry • 24 8.3 Stein Manifolds • 25 9 February 20 • 26 9.1 Stein Manifolds Continued • 26 9.2 Topological Properties of Kähler Manifolds • 26 9.3 Complex and Symplectic Structures on 4-Manifolds • 27 10 February 25 -
Double Groupoids and the Symplectic Category
JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2018009 c American Institute of Mathematical Sciences Volume 10, Number 2, June 2018 pp. 217{250 DOUBLE GROUPOIDS AND THE SYMPLECTIC CATEGORY Santiago Canez~ Department of Mathematics Northwestern University 2033 Sheridan Road Evanston, IL 60208-2730, USA (Communicated by Henrique Bursztyn) Abstract. We introduce the notion of a symplectic hopfoid, a \groupoid-like" object in the category of symplectic manifolds whose morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on a new realization of the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid- like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions. 1. Introduction. The symplectic category is the category-like structure whose objects are symplectic manifolds and morphisms are canonical relations, i.e. La- grangian submanifolds of products of symplectic manifolds. Although compositions in this \category" are only defined between certain morphisms, this concept has nonetheless proved to be useful for describing many constructions in symplectic ge- ometry in a more categorical manner; in particular, symplectic groupoids can be characterized as certain monoid objects in this category, and Hamiltonian actions can be described as actions of such monoids. -
A C1 Arnol'd-Liouville Theorem
A C1 Arnol’d-Liouville theorem Marie-Claude Arnaud, Jinxin Xue To cite this version: Marie-Claude Arnaud, Jinxin Xue. A C1 Arnol’d-Liouville theorem. 2016. hal-01422530v2 HAL Id: hal-01422530 https://hal-univ-avignon.archives-ouvertes.fr/hal-01422530v2 Preprint submitted on 16 Aug 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A C1 ARNOL’D-LIOUVILLE THEOREM MARIE-CLAUDE ARNAUD†, JINXIN XUE Abstract. In this paper, we prove a version of Arnol’d-Liouville theorem for C1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol’d- Liouville coordinates. We also explore various notions of C0 and Lipschitz integrability. Dedicated to Jean-Christophe Yoccoz 1. Introduction and Main Results. This article elaborates on the following question. Question. If a Hamiltonian system has enough commuting integrals1, can we precisely describe the Hamiltonian Dynamics, even in the case of non C2 integrals? When the integrals are C2, Arnol’d-Liouville Theorem (see [4]) gives such a dynamical description. -
Homotopy Type of Symplectomorphism Groups of S2 × S2 S´Ilvia Anjos
ISSN 1364-0380 (on line) 1465-3060 (printed) 195 Geometry & Topology Volume 6 (2002) 195–218 Published: 27 April 2002 Homotopy type of symplectomorphism groups of S2 × S2 S´ılvia Anjos Departamento de Matem´atica Instituto Superior T´ecnico, Lisbon, Portugal Email: [email protected] Abstract In this paper we discuss the topology of the symplectomorphism group of a product of two 2–dimensional spheres when the ratio of their areas lies in the interval (1,2]. More precisely we compute the homotopy type of this sym- plectomorphism group and we also show that the group contains two finite dimensional Lie groups generating the homotopy. A key step in this work is to calculate the mod 2 homology of the group of symplectomorphisms. Although this homology has a finite number of generators with respect to the Pontryagin product, it is unexpected large containing in particular a free noncommutative ring with 3 generators. AMS Classification numbers Primary: 57S05, 57R17 Secondary: 57T20, 57T25 Keywords: Symplectomorphism group, Pontryagin ring, homotopy equiva- lence Proposed:YashaEliashberg Received:1October2001 Seconded: TomaszMrowka,JohnMorgan Revised: 11March2002 c Geometry & Topology Publications 196 S´ılvia Anjos 1 Introduction In general symplectomorphism groups are thought to be intermediate objects between Lie groups and full groups of diffeomorphisms. Although very little is known about the topology of groups of diffeomorphisms, there are some cases when the corresponding symplectomorphism groups are more understandable. For example, nothing is known about the group of compactly supported dif- feomorphisms of R4 , but in 1985, Gromov showed in [4] that the group of compactly supported symplectomorphisms of R4 with its standard symplectic structure is contractible. -
An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron
An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron. Buchstaber, Victor M and Ray, Nigel 2008 MIMS EPrint: 2008.31 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Contemporary Mathematics An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron Victor M Buchstaber and Nigel Ray 1. An Invitation Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid-1990s began using the term toric topology to describe our widening interests in certain well-behaved actions of the torus. Little did we realise that, within seven years, a significant international conference would be planned with the subject as its theme, and delightful Japanese hospitality at its heart. When first asked to prepare this article, we fantasised about an authorita- tive and comprehensive survey; one that would lead readers carefully through the foothills above which the subject rises, and provide techniques for gaining sufficient height to glimpse its extensive mathematical vistas. All this, and more, would be illuminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale avail- able. Simultaneously, we understood that at least as valuable a service could be rendered to conference participants by an invitation to a wider mathematical au- dience - an invitation to savour the atmosphere and texture of the subject, to consider its geology and history in terms of selected examples and representative literature, to glimpse its exciting future through ongoing projects; and perhaps to locate favourite Osaka lectures within a novel conceptual framework. -
Margaret Dusa Mcduff Hon Dsc Warwick Oral Version
Margaret Dusa McDuff Hon DSc Warwick oral version Mr Vice Chancellor, Graduates, Graduands, Ladies and Gentlemen, We have with us today a mathematician whose work has opened a hugely fertile new branch of mathematics. She has brought symplectic geometry and topology to the attention of the mathematical world. Dusa McDuff is Professor of Mathematics at Barnard College in New York. Born in London, Dusa grew up in Edinburgh, where her father the influential biologist C.H. Waddington was Professor of Genetics. Dusa wanted to be a mathematician from an early age. She studied in Edinburgh then Cambridge. Her doctoral work was published in the top journal Annals of Mathematics and remains important today. Subsequently Dusa studied in Moscow with the great Russian mathematician Israel Gelfand. After various temporary posts, in 1976 she was appointed to a lectureship at Warwick. However in 1978 she moved to the State University of New York at Stony Brook. In 2007 she was appointed to the Kimmel chair at Barnard, sister college to Columbia University. Shortly after her move to the US, Dusa's research shifted towards symplectic geometry. This subject has its origins in mechanics and continues to be important in various branches of physics. In topology the objects are flabby, and in geometry they are rigid. Symplectic geometry sits in between. In the late 1970s symplectic geometry was beginning to develop in completely new directions, spurred by deep ideas introduced by Mikhail Gromov (one of the world's most creative living mathematicians). Over the years Dusa and her students have played a central role in developing this new field called symplectic topology. -
2006-2007 Graduate Studies in Mathematics Handbook
2006-2007 GRADUATE STUDIES IN MATHEMATICS HANDBOOK TABLE OF CONTENTS 1. INTRODUCTION .................................................................................................................... 1 2. DEPARTMENT OF MATHEMATICS ................................................................................. 2 3. THE GRADUATE PROGRAM.............................................................................................. 5 4. GRADUATE COURSES ......................................................................................................... 8 5. RESEARCH ACTIVITIES ................................................................................................... 24 6. ADMISSION REQUIREMENTS AND APPLICATION PROCEDURES ...................... 25 7. FEES AND FINANCIAL ASSISTANCE............................................................................. 25 8. OTHER INFORMATION..................................................................................................... 27 APPENDIX A: COMPREHENSIVE EXAMINATION SYLLABI ............................................ 31 APPENDIX B: APPLIED MATHEMATICS COMPREHENSIVE EXAMINATION SYLLABI ............................................................................................................................................................. 33 APPENDIX C: PH.D. DEGREES CONFERRED FROM 1994-2006........................................ 35 APPENDIX D: THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES........................................................................................................................................ -
Emissary | Spring 2021
Spring 2021 EMISSARY M a t h e m a t i c a lSc i e n c e sRe s e a r c hIn s t i t u t e www.msri.org Mathematical Problems in Fluid Dynamics Mihaela Ifrim, Daniel Tataru, and Igor Kukavica The exploration of the mathematical foundations of fluid dynamics began early on in human history. The study of the behavior of fluids dates back to Archimedes, who discovered that any body immersed in a liquid receives a vertical upward thrust, which is equal to the weight of the displaced liquid. Later, Leonardo Da Vinci was fascinated by turbulence, another key feature of fluid flows. But the first advances in the analysis of fluids date from the beginning of the eighteenth century with the birth of differential calculus, which revolutionized the mathematical understanding of the movement of bodies, solids, and fluids. The discovery of the governing equations for the motion of fluids goes back to Euler in 1757; further progress in the nineteenth century was due to Navier and later Stokes, who explored the role of viscosity. In the middle of the twenti- eth century, Kolmogorov’s theory of tur- bulence was another turning point, as it set future directions in the exploration of fluids. More complex geophysical models incorporating temperature, salinity, and ro- tation appeared subsequently, and they play a role in weather prediction and climate modeling. Nowadays, the field of mathematical fluid dynamics is one of the key areas of partial differential equations and has been the fo- cus of extensive research over the years. -
Mathematicians I Have Known
Mathematicians I have known Michael Atiyah http://www.maths.ed.ac.uk/~aar/atiyahpg Trinity Mathematical Society Cambridge 3rd February 20 ! P1 The Michael and Lily Atiyah /ortrait 0allery 1ames Clerk Ma'well 2uilding, 3niversity o# *dinburgh The "ortraits of mathematicians dis"layed in this collection have been "ersonally selected by us. They have been chosen for many di##erent reasons, but all have been involved in our mathematical lives in one way or another; many of the individual te'ts to the gallery "ortraits e'"lain how they are related to us. First% there are famous names from the "ast ( starting with Archimedes ( who have built the great edifice of mathematics which we inhabit$ This early list could have been more numerous, but it has been restricted to those whose style is most a""ealing to us. )e't there are the many teachers, both in Edinburgh and in Cambridge% who taught us at various stages% and who directly influenced our careers. The bulk of the "ortraits are those o# our contemporaries, including some close collaborators and many Fields Medallists. +ily has a special interest in women mathematicians: they are well re"resented% both "ast and "resent$ Finally we come to the ne't generation, our students$ -f course% many of the categories overla"% with students later becoming collaborators and #riends. It was hardest to kee" the overall number down to seventy, to #it the gallery constraints! P2 4 Classical P3 Leonhard Euler 2asel 505 ( St$ /etersburg 563 The most proli#ic mathematician of any period$ His collected works in more than 53 volumes are still in the course of publication.