Lie algebroids, Poisson manifolds and Jacobi structures

This is the website for the Lie algebroids, Poisson manifolds and Jacobi structures working seminar, organised by Carlos Zapata-Carratalá within the Edinburgh Mathematical Physics Group.

Originating in the mathematics of classical mechanics, Poisson geometry has become a vibrant branch of pure mathematics in the last few decades, with deep connections to groupoid theory, non-commutative geometry and integrable systems. In this minicourse we aim to give a broad introduction to the subject to, hopefully, spark interest for further study in those who attend the lectures.

The lectures take place over a 2-hour period with a 5-minute break and time for questions, encouraged both at the end and during the lectures. Due to the COVID-19 pandemic we have moved to online seminars held over Zoom. To join a lecture follow this link. The Zoom meeting will typically open around 15 minutes before the scheduled times. You may click on the title of past lectures to access the lecture notes.

Schedule of seminars

1. Friday, 24 January 2020, 15:30-16:30
   Seminar Room 5.02, ICMS, Bayes Centre
   Historical introduction and organisational meeting
   Carlos Zapata-Carratalá
2. Friday, 07 February 2020, 16:00-18:00
   Lecture Theatre 5.10, ICMS, Bayes Centre
   Poisson Geometry
   Carlos Zapata-Carratalá
3. Friday, 14 February 2020, 16:00-18:00
   Lecture Theatre 5.10, ICMS, Bayes Centre
   Lie Groupoids
   Carlos Zapata-Carratalá
4. Friday, 21 February 2020, 16:00-18:00
   Seminar Room 5.02, ICMS, Bayes Centre
   Lie Algebroids
   Carlos Zapata-Carratalá
5. Friday, 28 February 2020, 16:00-18:00
   Lecture Theatre 5.10, ICMS, Bayes Centre
   Differential Operators (I)
   Sue Sierra
6. Friday, 06 March 2020, 15:00-17:00
   Lecture Theatre 5.10, ICMS, Bayes Centre
   Differential Operators (II)
   Carlos Zapata-Carratalá
7. Friday, 13 March 2020, 16:00-18:00
   Seminar Room 5.02, ICMS, Bayes Centre
   Local Lie Algebras
   Carlos Zapata-Carratalá
8. Friday, 8 May 2020, 16:00-18:00
   Zoom
   Jacobi Geometry I
   Carlos Zapata-Carratalá
9. Friday, 15 May 2020, 16:00-18:00
   Zoom
   Jacobi Geometry II
   Carlos Zapata-Carratalá
10. Friday, 22 May 2020, 16:00-18:00
    Zoom
    Dirac Geometry I
    Carlos Zapata-Carratalá
11. Friday, 29 May 2020, 16:00-18:00
    Zoom
    Dirac Geometry II
    Carlos Zapata-Carratalá
    Spinors and Dirac Structures I
    José Figueroa-O'Farrill
12. Friday, 5 June 2020, 16:00-18:00
    Zoom
    Spinors and Dirac Structures II
    José Figueroa-O'Farrill
13. Friday, 12 June 2020, 16:00-18:00
    Zoom
    Spinors and Dirac Structures III
    José Figueroa-O'Farrill
14. Friday, 19 June 2020, 16:00-18:00
    Zoom
    Shifted Poisson Structures
    Jon Pridham

References

• Lectures on Poisson Geometry
  R. Fernandes, I. Marcut
  Standard modern reference for Poisson manifolds.
• General Theory of Lie Groupoids and Lie Algebroids
  K. Mackenzie
  Standard reference on Lie groupoids and Lie algebroids.
• Groupoids: unifying internal and external symmetry
  A. Weinstein
  A conceptual discussion on Lie groupoids.
• A Landscape of Hamiltonian Phase Spaces (Thesis)
  C. Zapata
  Chapter 2 contains a concise introduction to most of the topics covered in the minicourse.
• Generalized Complex Geometry (Thesis)
  M. Gualtieri
  Chapters 2 and 3 cover Courant algebroids and Dirac structures.
• Deformations of coisotropic submanifolds in Jacobi manifolds (Thesis)
  A. Tortorella
  Chapters 1 and 2 introduce the general formalism of line bundles and Jacobi structures. Section 4.1 discusses Contact manifolds in some detail.
• Brackets, forms and invariant functionals
  N. Hitchin
  The first section motivates the Courant bracket in a very natural way using spinors.
• Pure Spinors on Lie Groups
  A. Alekseev, H. Bursztyn, E. Meinrenken
  Sections 1 and 2 cover linear Dirac structures as well as the spinor approach to Courant algebroids.
• An outline of Shifted Poisson structures
  J. Pridham
• Poisson-Lie Structures and Shifted Poisson Structures
  P. Safronov
• Shifted Poisson Structures on Differentiable Stacks
  F. Bonechi, N. Ciccoli, C. Laurent-Gengoux, P. Xu
• Smooth manifolds and observables
  J. Nestruev
• A primer of algebraic D-modules
  S.C. Coutinho