水木社区手机版
- 主题:seminar 20210907~30 Geometry and Dynamics of Singular Sympl
- https://web.mat.upc.edu/eva.miranda/coursHenan.htm
Geometry and Dynamics of Singular Symplectic manifolds
Henan University by Eva Miranda
NEW CONNECTION DATA!
The new Zoom link for lecture 2 and on:
New Zoom ID: 815 7127 3363
Code: 123456
Problem sessions by Joaquim Brugu?s joaquim.brugues@upc.edu and Pau Mir pau.mir.garcia@upc.edu
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 Some classical problems for b^m-symplectic and b^m-contact manifolds: The (singular) Weinstein conjecture. Connection to escape orbits in Celestial Mechanics.
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: Finale: Open problems including Arnold conjecture and Floer homology of Singular Symplectic Manifolds.
Material:
Slides day 1, annotated slides day 1, recordings from lecture 1 available NEW!
Link: https://zoom.com.cn/rec/share/HIjqT2fTv1xNQTn2maZ_yZXGWROfqUer4n8D_K11S4zXwIiw3fxESA4sAYOyr2HR.3kpmBm8U-Crf2hy-
Passcode: $4P2gh2V
Slides day 2, annotated slides day 2, recordings from lecture 2 available NEW!
https://us02web.zoom.us/rec/share/g5BKbJ8ocAnuUJRqy2qvYD01VeEUWmcTrBSInq7Rpa9FJYDHXGfl5b40-Znu3_-V.Yv9HUZDrN4DIAKbq
(passcode: KO8i^i77)
Slides day 3
Notes of the course (this file will be updated regularly).
List of Problems.
Poster of the course.
Videos of the course (soon on my youtube channel):
Link: https://zoom.com.cn/rec/share/HIjqT2fTv1xNQTn2maZ_yZXGWROfqUer4n8D_K11S4zXwIiw3fxESA4sAYOyr2HR.3kpmBm8U-Crf2hy-
Passcode: $4P2gh2V
https://us02web.zoom.us/rec/share/g5BKbJ8ocAnuUJRqy2qvYD01VeEUWmcTrBSInq7Rpa9FJYDHXGfl5b40-Znu3_-V.Yv9HUZDrN4DIAKbq
(passcode: KO8i^i77)
--
修改:vinbo FROM 202.121.181.*
FROM 202.121.181.*