Geometry and Dynamics of Singular
Symplectic Manifolds
IHP,
Paris, NovemberDecember 2017, Salle 314
This course is done in the framework of Eva
Miranda's Chaire
d'Excellence de la Fondation Sciences Mathématiques de Paris
Poster/affiche
Official webpage:
https://www.sciencesmathsparis.fr/fr/lecoursdevamiranda933.htm
Summary: bCalculus was introduced by Richard Melrose when
considering pseudodifferential operators on manifolds with boundary. Later on,
Ryszard Nest and Boris Tsygan applied these ideas to study the deformation
quantization of symplectic manifolds with boundary.
The purpose of this minicourse is to unravel the geometrical structures (bsymplectic
structures) behind this picture and describe some applications to Dynamical
systems. bSymplectic manifolds are Poisson manifolds which are symplectic away
from an hypersurface and satisfy some transversality condition. bSymplectic
manifolds lie "close enough" to the symplectic category and indeed their study
can be addressed using an "extended" De Rham complex. In particular many
peculiarities from Symplectic manifolds are shared with bsymplectic manifolds.
Using these ideas, we will study normal form theorems, actionangle theorems,
toric actions and applications to KAM theory. At the end of the minicourse we
present other singular symplectic structures such as folded symplectic
structures and b^msymplectic structures (for which the transversality condition
is relaxed) and explain how they are related to bsymplectic and symplectic
structures.
We will give a general overview of the theory using some examples in celestial
mechanics as leitmotiv. For some of them (like double collision), we can even
construct b^msymplectic structures and mfolded structures. This apparent "duality"
will be used as an excuse to closely explore the relation between the $b^m$symplectic
category with the symplectic and folded symplectic category. This relation
depends surprisingly on the parity of m and is given by a desingularization
procedure called deblogging. Time permitting, several applications of deblogging
to dynamics and quantization will be presented.
Syllabus/Scheme of the
lectures

Lecture 1: Introduction. Poisson manifolds: First Examples.

Lecture 2: Classical examples. The language of bivector
fields. Multivector calculus and the Schouten bracket. Symplectic foliations.

Lecture 3: Weinstein's splitting theorem and normal forms.
Conn's linearization theorem. Poisson Cohomology computation kit.

Lecture 4: Poisson cohomology and integrable systems on
Poisson manifolds.

Lecture 5: Part I:
Proof of existence of
actionangle coordinates in Poisson Geometry. Part II:
Introduction to bPoisson Geometry.

Lecture 6: Part I:
Examples in bPoisson Geometry.
bDarboux theorem. Modular vector fields in Poisson Geometry. Part II:
Codimension one symplectic
foliations and unimodular Poisson structures.

Lecture 7: Part I
Poisson Geometry of the critical
hypersurface of a bPoisson manifold. Part II.
Changing the glasses: A dual
language for bPoisson manifolds. bSymplectic structures. bCohomology and
Poisson Cohomology.

Lecture 8: The
path method in bSymplectic Geometry: Applications to classification
theorems in bsymplectic geometry (including Delzant
theorem and actionangle coordinates for $b$symplectic manifolds).

Lecture 9: Deblogging: The magic stick. How to convert a $b^m$symplectic
manifold into a (folded) symplectic one.

Lecture 10: Applications of Deblogging to
Dynamics and
Quantization. Open problems (and
open problems for the Working group).
Material:
Videos of the course:
Available in the webpage of the course at FSMP:https://www.sciencesmathsparis.fr/fr/lecoursdevamiranda933.htm
Bibliography:
 [BDMOP]
R. Braddell, A. Delshams, E. Miranda, C. Oms and A. Planas,
An invitation to Singular Symplectic Geometry ,
arXiv:1705.03846, accepted at International Journal of Geometric Methods in
Modern Physics, 2017.
 [CGP] A. Cannas, V. Guillemin, A.R. Pires, Symplectic Origami , International
Mathematics Research Notices, no.18, pp 42524293, 2011.
 [DKM] A. Delshams, A. Kiesenhofer, E. Miranda, Examples of integrable
and nonintegrable systems on singular symplectic manifolds,
J. Geom. Phys.
115 (2017), 89–97.
 [DZ]
Dufour, JeanPaul;
Zung, Nguyen Tien Poisson structures and their
normal forms.
Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005.
xvi+321 pp. ISBN: 9783764373344; 3764373342
 [DKRS] A. Delshams, V. Kaloshin, A de la Rosa, T. M.Seara, Global
instability in the elliptic restricted three body problem, arXiv:1501.01214.
 [Du] J.J. Duistermaat, On global actionangle coordinates. Comm. Pure
Appl. Math. 33 (1980), no. 6, 687706.
 [GMP1] V. Guillemin, E. Miranda, and A. Pires, Codimension one
symplectic foliations and regular Poisson structures. Bull. Braz. Math. Soc.
(N.S.), 42(4):607623, 2011.
 [GMP2] V. Guillemin, E. Miranda, and A. Pires,
Symplectic and Poisson geometry on
bmanifolds. Adv. Math. 264 (2014), 864896.
 [GMPS1] V. Guillemin, E. Miranda, A. R. Pires and G. Scott,
Toric actions on bsymplectic
manifolds, Int Math Res Notices Int Math Res Notices (2015) 2015 (14):
58185848.
 [GMPS2] V. Guillemin, E. Miranda, A. Pires, and G. Scott.
Convexity for Hamiltonian torus
actions on bsymplectic manifolds,
Math. Res. Lett.
24 (2017), no. 2, 363–377.
 [GMW1] V. Guillemin, E. Miranda, J. Weitsman,
Desingularizing b^msymplectic
structures,
 [GMW2] V. Guillemin, E. Miranda, J. Weitsman, On geometric quantization
of bsymplectic manifolds ,arXiv:1608.08667.
 [LMV] C. LaurentGengoux, E. Miranda and P. Vanhaecke,
Actionangle coordinates for integrable systems on
Poisson manifolds.
Int. Math. Res. Not. IMRN
2011, no. 8, 1839–1869.
 [LPV]
LaurentGengoux, Camille;
Pichereau, Anne;
Vanhaecke, Pol Poisson structures.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], 347. Springer, Heidelberg, 2013. xxiv+461
pp. ISBN: 9783642310898.
 [MG]R. McGehee, Singularities in classical celestial mechanics.
Proceedings of the International Congress of Mathematicians (Helsinki,
1978), pp. 827834, Acad. Sci. Fennica, Helsinki, 1980.
 [W] A. Weinstein, The local structure of Poisson manifolds., J.
Differential Geom. 18 (1983), no. 3, 523557.
Complementary material:

Albert Einstein, Zum
Quantensatz von Sommerfeld und Epstein, Deutsche Physikalische
Gesellschaft, Verhandlungen 19, 8292 (1917), translation in Portuguese
published at Revista Brasileira de Ensino de Fisica, v. 27, n. 1, p. 103 
107, (2005).

KosmannSchwarzbach, Yvette Les crochets de Poisson,
de la mécanique céleste à la mécanique quantique, SiméonDenis
Poisson, 369–401,
Hist. Math. Sci. Phys., Ed. Éc. Polytech., Palaiseau, 2013.

KosmannSchwarzbach, Yvette La géométrie de Poisson,
création du XXe siècle. (French) [Poisson geometry, a twentiethcentury
creation] SiméonDenis Poisson, 129–172,
Hist. Math. Sci. Phys., Ed. Éc. Polytech., Palaiseau, 2013.
 Link to the Catalogue of the
Exhibition "SiméonDenis Poisson Mathematics at the service of
Science" which took place in Paris and UrbanaChampaign in 2014 courtesy
of Yvette KosmannSchwarzbach, Version en Français ici
Catalogue merci à Yvette KosmannSchwarzbach.
Related activities
of the Chair:
 Working group on bsymplectic geometry and Celestial
Mechanics (every Thursday during the course from 4 to 5pm at Salle 314 at
Institut Henri Poincaré we discuss about applications of bSymplectic
Geometry to Celestial Mechanics). We are: Roisin Braddell, Eva Miranda,
Cédric Oms, Michael Orieux, Wang Qun and anybody who wants to join. Join us!

Closing conference of the Chair.

Working group on
the nbody problem (on Mondays at Observatoire de Paris)

Séminaire de Géométrie Hamiltonienne (on Fridays at Jussieu).