Muutke küpsiste eelistusi

Lectures on Poisson Geometry [Pehme köide]

  • Formaat: Paperback / softback, 479 pages, kaal: 894 g
  • Sari: Graduate Studies in Mathematics
  • Ilmumisaeg: 30-Dec-2021
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470466678
  • ISBN-13: 9781470466671
Teised raamatud teemal:
  • Formaat: Paperback / softback, 479 pages, kaal: 894 g
  • Sari: Graduate Studies in Mathematics
  • Ilmumisaeg: 30-Dec-2021
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470466678
  • ISBN-13: 9781470466671
Teised raamatud teemal:
Each refining their lecture notes for graduate courses and seminars for many years, Crainic, Fernandes, and Marcut have melded them to provide an introduction to Poisson geometry that contains more material than a single-semester course can cover. In sections on basic concepts, Poisson geometry around leaves, global aspects, and symplectic groupoids, they consider such topics as Poisson brackets, symplectic leaves and the symplectic foliation, submanifolds in Poisson geometry, contra-variant geometry and connections, and a crash course on Lie groupoids. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)

Arvustused

This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way."" Alan Weinstein, University of California at Berkeley

""This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study."" Eckhard Meinrenken, University of Toronto

Preface xiii
List of Notations and Symbols xvii
List of Conventions xix
Part 1 Basic Concepts
Chapter 1 Poisson Brackets
3(20)
1.1 Poisson brackets
3(2)
1.2 Orbits
5(1)
1.3 Poisson and Hamiltonian diffeomorphisms
6(2)
1.4 Examples
8(6)
1.5 Poisson actions and quotients
14(5)
Problems
19(4)
Chapter 2 Poisson Bivectors
23(20)
2.1 The point of view of bivectors
23(5)
2.2 A slight twist: 70
28(2)
2.3 Poisson maps and bivector fields
30(1)
2.4 Examples
31(9)
Problems
40(3)
Chapter 3 Local Structure of Poisson Manifolds
43(16)
3.1 The Weinstein Splitting Theorem
43(3)
3.2 Regular points
46(1)
3.3 Singular points
47(3)
3.4 The isotropy Lie algebra
50(3)
3.5 Linearization of Poisson structures
53(2)
Problems
55(4)
Notes and References for Part 1
59(4)
Part 2 Poisson Geometry Around Leaves
Chapter 4 Symplectic Leaves and the Symplectic Foliation
63(22)
4.1 The symplectic foliation
63(6)
4.2 Regular Poisson structures
69(4)
4.3 More examples of symplectic foliations
73(3)
4.4 The coupling construction
76(5)
Problems
81(4)
Chapter 5 Poisson Transversals
85(16)
5.1 Slices and Poisson transversals
85(6)
5.2 The transverse Poisson structure to a leaf
91(4)
5.3 Poisson maps and Poisson transversals
95(2)
Problems
97(4)
Chapter 6 Symplectic Realizations
101(32)
6.1 Definition
101(2)
6.2 Examples
103(8)
6.3 Symplectic realizations of linear Poisson structures
111(6)
6.4 Libermann's Theorem and dual pairs
117(7)
6.5 Local existence
124(5)
Problems
129(4)
Chapter 7 Dirac Geometry
133(24)
7.1 Constant Dirac structures
134(3)
7.2 Dirac structures
137(5)
7.3 Pullbacks of Dirac structures
142(3)
7.4 Pushforwards of Dirac structures
145(4)
7.5 Gauge equivalences
149(4)
Problems
153(4)
Chapter 8 Submanifolds in Poisson Geometry
157(36)
8.1 Poisson submanifolds
157(8)
8.2 Poisson-Dirac submanifolds
165(4)
8.3 Coregular Poisson-Dirac submanifolds
169(3)
8.4 Coisotropic submanifolds
172(8)
8.5 Example: Fixed point sets
180(4)
8.6 Pre-Poisson submanifolds
184(5)
Problems
189(4)
Notes and References for Part 2
193(4)
Part 3 Global Aspects
Chapter 9 Poisson Cohomology
197(26)
9.1 The cotangent Lie algebroid
197(3)
9.2 The Poisson differential and Poisson cohomology
200(1)
9.3 Low degrees
201(5)
9.4 Shadows of Poisson cohomology
206(7)
9.5 The cohomological obstruction to linearization
213(5)
Problems
218(5)
Chapter 10 Poisson Homotopy
223(30)
10.1 Cotangent paths
223(2)
10.2 Cotangent maps
225(3)
10.3 Integration and the contravariant Stokes Theorem
228(3)
10.4 Cotangent path-homotopy
231(6)
10.5 Poisson homotopy and homology groups
237(8)
10.6 Variation of symplectic area
245(4)
Problems
249(4)
Chapter 11 Contravariant Geometry and Connections
253(24)
11.1 Contravariant connections on vector bundles
253(3)
11.2 Parallel transport along cotangent paths
256(4)
11.3 Flat contravariant connections
260(5)
11.4 Geodesics for contravariant connections
265(4)
11.5 Existence of symplectic realizations
269(4)
Problems
273(4)
Notes and References for Part 3
277(6)
Part 4 Symplectic Groupoids
Chapter 12 Complete Symplectic Realizations
283(30)
12.1 The infinitesimal action
284(2)
12.2 Case study: Linear Poisson structures
286(2)
12.3 Case study: The zero Poisson structure
288(2)
12.4 Case study: Nondegenerate Poisson structures
290(3)
12.5 Completeness
293(6)
12.6 The Poisson homotopy groupoid
299(3)
12.7 Lagrangian fibrations
302(7)
Problems
309(4)
Chapter 13 A Crash Course on Lie Groupoids
313(48)
13.1 Lie groupoids
313(3)
13.2 Lie groupoids: Examples and basic constructions
316(10)
13.3 The Lie algebroid of a Lie groupoid
326(4)
13.4 Lie algebroids: Examples and basic constructions
330(9)
13.5 Duals of Lie algebroids
339(6)
13.6 The Lie philosophy
345(3)
13.7 The non-Hausdorff setting
348(9)
Problems
357(4)
Chapter 14 Symplectic Groupoids
361(58)
14.1 Symplectic groupoids and Poisson structures
361(7)
14.2 Examples
368(9)
14.3 Integrability of Poisson structures I
377(11)
14.4 Symplectic groupoid actions
388(5)
14.5 Hausdorffness issues
393(2)
14.6 The Poisson homotopy groupoid
395(10)
14.7 Morita equivalence
405(3)
14.8 Integrability of Poisson structures II
408(6)
Problems
414(5)
Notes and References for Part 4
419(6)
Part 5 Appendices
Appendix A. Lie Groups
425(12)
A.1 Lie groups
425(3)
A.2 Lie group actions
428(5)
A.3 Time-dependent vector fields
433(4)
Appendix B. Symplectic Structures
437(10)
B.1 Symplectic forms
437(4)
B.2 Symplectic and Hamiltonian actions
441(6)
Appendix C. Foliations
447(12)
C.1 Regular foliations
447(5)
C.2 Foliated differential forms
452(2)
C.3 Singular foliations
454(5)
Appendix D. Groupoids: Conventions and Choices
459(4)
Bibliography 463(10)
Index 473
Marius Crainic, Utrecht University, The Netherlands.

Rui Loja Fernandes, University of Illinois at Urbana-Champaign, IL.

Ioan Marcut, Radboud University, Nijmegen, The Netherlands.