Geometry and Dynamics of Singular 
Symplectic manifolds
Henan University by Eva Miranda
 
Public 
Lecture takes place on October 11 at 13h CET, 19h Kaifeng time
 
Connecting data for Public 
Lecture:
Zoom Id: 5673065241
password: 123456
Undecidability of certain 
fluid paths, the Navier-Stokes problem and 29000 rubber ducks lost in the ocean
 

 
 
This course is part of the 
Overseas Distinguished Lectureship Program at the University of Henan in Kaifeng
NEW CONNECTION DATA!
	- 
	The new Zoom link for lecture 2 and 
	on: 
	
New Zoom ID: 815 7127 3363
	Code: 123456
	 
	
	
	 
 


Figure on the top: Image 
of the moment map of a b-torus (courtesy of Pablo Nicolás, a student of the 
course). Observe that this b-moment map is circle valued. Figure on the bottom 
representation of a b-map on a b-sphere.
Summary: b-Calculus 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 (b-symplectic 
structures) behind this picture and describe some applications to Dynamical 
systems. b-Symplectic manifolds are Poisson manifolds which are symplectic away 
from an hypersurface and satisfy some transversality condition. b-Symplectic 
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 b-symplectic manifolds. 
Using these ideas, we will study normal form theorems, action-angle 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^m-symplectic structures (for which the transversality condition 
is relaxed) and explain how they are related to b-symplectic 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^m-symplectic structures and m-folded 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
	The planning of the lectures would be the following one:
	
 
	Title: Geometry and Dynamics of Singular Symplectic Manifolds 
	Summary: We will describe a novel geometrical approach to classical problems 
	in Celestial Mechanics concerning collisions. The upshot of our methods is 
	that the singularities (collisions, infinity line) are included in the 
	geometrical techniques(as b-symplectic manifolds, b-contact manifolds). We 
	will focus on the geometry and Dynamics of these manifolds and describe 
	several techniques such as desingularization, normal forms, action-angle 
	coordinates and perturbation theory used in this study.
 
	
 
	Planning with description of contents per day.
	
 
	September 7 Overture: Introduction to the course. Basic definitions 
	in Symplectic Geometry and motivation for b-symplectic geometry. B-symplectic 
	manifolds as Poisson manifolds.
 
	September 9 Melrose language of b-forms. b-symplectic forms on 
	b-Poisson manifolds. The geometry of the critical set. More degenerate forms 
	b^m-symplectic forms and b^m contact forms. Desingularization of b^m-forms.
 
	September 14  The path method for b^m-symplectic structures. Local 
	normal form (b^m-Darboux theorem) and extension theorems. b^m-Structures to 
	the test: Examples in Fluid Dynamics and Celestial Mechanics. The b-symplectic 
	and b-contact geometry of the restricted three body problem and of Beltrami 
	fields. Application: Finding periodic orbits for trajectories of a satellite 
	in the restricted three body problem.
 
	September 16 Exercise session
 
	September 21 (POSTPONED DUE TO NATIONAL FESTIVITY IN CHINA)
 
	September 23  More symmetries: Toric actions, action-angle 
	coordinates and Integrable systems on b^m-symplectic manifolds. Applications: 
	Perturbations of integrable systems and KAM theory.
 
	September 28: Exercise session
 
	September 30:  Some classical problems for b^m-symplectic and b^m-contact 
	manifolds. Examples in Celestial Mechanics. Periodic Orbits. The (singular) Weinstein conjecture. 
October 5: Finale: 
	A magic mirror between singular contact geometry and singular Fluid Dynamics. 
	Applying Uhlenbeck's results to find escape orbits in Celestial Mechanics. Open problems.
 

Material:
	
		
			
			
				(passcode: KO8i^i77)
		
	
Videos of the course :
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 4252-4293, 2011.
- [CM] R. Cardona and E. Miranda, Integrable systems on singular 
	symplectic manifolds: from local to global, IMRN to appear, 2021,
	https://arxiv.org/abs/2007.10314 
- [DKM] A. Delshams, A. Kiesenhofer, E. Miranda, Examples of integrable 
	and non-integrable systems on singular symplectic manifolds,
	
	J. Geom. Phys.
	
	115 (2017), 89–97.
- [DZ]
	
	Dufour, Jean-Paul;
	
	Zung, Nguyen Tien Poisson structures and their 
	normal forms.
	
	Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. 
	xvi+321 pp. ISBN: 978-3-7643-7334-4; 3-7643-7334-2
- [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 action-angle coordinates. Comm. Pure 
	Appl. Math. 33 (1980), no. 6, 687-706.
- [GMP1] V. Guillemin, E. Miranda, and A. Pires, Codimension one 
	symplectic foliations and regular Poisson structures. Bull. Braz. Math. Soc. 
	(N.S.), 42(4):607-623, 2011.
- [GMP2] V. Guillemin, E. Miranda, and A. Pires,
	Symplectic and Poisson geometry on 
	b-manifolds. Adv. Math. 264 (2014), 864-896.
- [GMPS1] V. Guillemin, E. Miranda, A. R. Pires and G. Scott,
	Toric actions on b-symplectic 
	manifolds, Int Math Res Notices Int Math Res Notices (2015) 2015 (14): 
	5818-5848.
- [GMPS2] V. Guillemin, E. Miranda, A. Pires, and G. Scott.
	Convexity for Hamiltonian torus 
	actions on b-symplectic manifolds,
	
	Math. Res. Lett.
	
	24 (2017), no. 2, 363–377. 
- [GMW1] V. Guillemin, E. Miranda, J. Weitsman,
	Desingularizing b^m-symplectic 
	structures,
	
	
- [GMW2] V. Guillemin, E. Miranda, J. Weitsman, On geometric quantization 
	of b-symplectic manifolds, Adv in Math 2018.
- [LMV] C. Laurent-Gengoux, E. Miranda and P. Vanhaecke,
	Action-angle coordinates for integrable systems on 
	Poisson manifolds.
	
	Int. Math. Res. Not. IMRN
	
	2011, no. 8, 1839–1869.
- [LPV]
	
	Laurent-Gengoux, Camille;
	
	Pichereau, Anne;
	
	Vanhaecke, Pol Poisson structures.
	
	Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of 
	Mathematical Sciences], 347. Springer, Heidelberg, 2013. xxiv+461 
	pp. ISBN: 978-3-642-31089-8.
- [MG]R. McGehee, Singularities in classical celestial mechanics. 
	Proceedings of the International Congress of Mathematicians (Helsinki, 
	1978), pp. 827-834, Acad. Sci. Fennica, Helsinki, 1980. 
- [MO1] E. Miranda and C. Oms,
	
	The Singular Weinstein conjecture, Advances in Math, 2021, Golden Open 
	access
- [MO2] E. Miranda and C. Oms, The geometry and topology of contact 
	structures with singularities, 
	https://arxiv.org/abs/1806.05638 
- [MP] E. Miranda and A. Planas, Action-angle coordinates and KAM for 
	singular symplectic manifolds, monograph 100 pages, 2021.
- [W] A. Weinstein, The local structure of Poisson manifolds., J. 
	Differential Geom. 18 (1983), no. 3, 523-557.