# Math 595 – Symplectic and Poisson Geometry – Fall 2015

This course is an introduction to Symplectic and Poisson geometry. Poisson geometry is the study of a manifold equipped with a Poisson bracket. In symplectic geometry one imposes a non-degeneracy condition on the Poisson bracket. The roots of these geometries lie in Classical Mechanics, but they became an independent field of study in the 1970’s. Nowadays, there is an impressive body of results with beautiful connections with many other areas of mathematics. Students taking this course are assumed to know differential geometry at the level of Math 518 – Differentiable Manifolds.

Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Mon and Wed 11:00 AM (or by appointment);
Class meets: TTh 11:00-12:20 PM, 441 Altgeld Hall;
Prerequisites: Math 518 or equivalent.

## Syllabus:

• Foundational aspects of Symplectic and Poisson Geometry. Symplectic Linear Algebra; Symplectic forms and Poisson brackets, Dirac structures. Symplectomorphism and Poisson maps, Hamiltonian vector fields. Submanifolds: Lagrangian and coisotropic submanifolds, symplectic submanifolds and Poisson transversals. Quotients and Reduction.
• Symmetry and reduction. Symplectic and Poisson actions. Quotients. Moment maps and symplectic quotients. Symplectic Toric manifolds. Delzant Theorem.
• Local aspects. Darboux-Weinstein Theorems. Tubular Neighborhood Theorems. Conn’s linearization Theorem. Normal forms around symplectic leaves.
• Global Aspects. Symplectic realizations. Integrability and Lie groupoids. Gromov’s Nonsqueezing Theorem.

## Textbooks:

• A. Cannas da Silva, Lectures on Symplectic Geometry, LNM 1764, Springer-Verlag, Berlin, 2001.
• A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, RI, 1999.
• J.-P. Dufour and N.T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, Vol. 242, BirkhÃ¤user, Basel, 2005.
• D. McDuff and D. Salamon, Introduction to Symplectic Toplogy, Oxford University Press, New York, 1998.

I am writing some lectures notes in Poisson geometry, with Ioan Marcut, which may also be useful.

• Expository Paper Students will be required to write (in LaTeX) and present a paper. The papers will be due by the last week of classes. Presentations will be in class, during the last week of classes. Some possible topics:
• Coisotropic/isotropic tubular neighborhood theorem
• Convexity theorem: proof and non-abelian version
• Fix points and Arnold conjecture
• Symplectic groupoids and Poisson structures
• Duistermaat-Heckman Theorem
• Symplectic realizations of Poisson manifolds
• Groups of symplectomorphism and flux homomorphism
• Linearization of Poisson structures
• Symplectic toric manifolds and toric varieties
• Problem Sessions There will be problem sessions which will be based in 6 homework assignments.

## Homework Assignments and Sections covered so far:

• Homework #1: Homeworks 1 and 2 of the book by A. Cannas da Silva
• Homework #2: Solve the following set of problems on the Schouten bracket.
• Homework #3: Homeworks 19 and 20 of the book by A. Cannas da Silva.
• Homework #4: Homeworks 8, 9 and 10 of the book by A. Cannas da Silva.

Material covered so far:

• Week 1: The nature of Symplectic Geometry: Symplectic vs Riemannian geometry. Poisson vs Symplectic Geometry. Symplectic linear algebra: symplectic vector spaces; symplectic linear maps; canonical form; isotropic/coisotropic/Lagrangian/symplectic subspaces. Symplectic manifolds: definition and first examples; Darboux’s Theorem: constructive proof. (CdS: 1.1-2.4; McDS: 1, 2.1-2.3)
• Week 2: Local vs Global invariants: Liouville volume form; cohomological obstructions; Submanifolds: isotropic/coisotropic/Lagrangian/symplectic submanifolds. Lagrangian submanifolds of the cotangent bundle; Lagrangian submanifolds and symplectomorphisms. Hamiltonian dynamics and Poisson brackets. (CdS: 3.1-3.4, 18.1-18.3; McDS: 3.1)
• Week 3: Poisson manifolds: Poisson maps, hamiltonian vector fields, Poisson vector fields; linear and quadratic Poisson structures.Contravariant calculus: multivector fields and multiderivations; Lie derivative and Schouten bracket. Regular Poisson structures and symplectic foliations. (CdSWe 3.1-4.2)
• Week 4: Non-regular Poisson manifolds: Weinstein splitting theorem, symplectic foliation, Poisson submanifolds, Casimir functions; Poisson cohomology: Schouten bracket vs contravariant differential, intrepertation of cohomology in low-degrees, fundamental class, modular class. (CdSWe 4.3-4.6, 3.6, 6.6)
• Week 5: Local normal forms: Moser’s Method and symplectic isotopies, Darboux’s Theorem, Symplectic Tubular Neighborhood Theorem, Lagrangian Tubular Neighborhood Theorem; Applications to fixed points and Arnold Conjecture; Conn’s Linearization Theorem. (McDS: 3.2, 3.3; CdS 7.1-9.4; CdSWe 5)
• Week 6: Symmetries and actions: actions by symplectomorphisms and by Poisson diffeomorphisms; Poisson quotients and reduction of Hamiltonian systems; Moment maps, Hamiltonian G-spaces, Hamiltonian quotients: the Meyer-Marsden-Weinstein reduction theorem; examples: Fubini-Study form on complex projective space, cotangent bundles. (McDS 5.1-5.3, CdS 23-24)
• Week 7: Reduction and Integrable Systems: Coajoint orbits and reduction at non-zero levels: orbit reduction and point reduction, shifting tricks. Lagrangian fibrations: integrable systems and reduction of hamiltinonian systems; integral affine structures; local normal form, action-angle variables and Arnold-Liouville theorem; global invariants: monodromy and Lagrangian Chern class. (McDS 1.1, CdS 18.1-18.4, J.J. Duistermaat paper)
• Week 8: Symplectic toric manifolds: hamiltonian torus actions, the Atiyah-Guillemin-Sternberg convexity theorem, effective hamiltonian torus actions; Delzant polytopes, Delzant Theorem, Delzant construction. (CdS 28.1-29.3)
• Week 9: Almost complex structures: complex vector spaces, compatible almost structures and metrics, applications of compatible almost complex structures; integrability and Nijenhuis torsion, Dolbeault complex, Kahler structures. (CdS 12.1-15.3)
• Week 10: Contact structures: contact forms, contact structures, Reeb vector field, symplectization. General constructions: 3-sphere, hypersurfaces transverse to Liouville vector fields, unit cosphere bundles. Gray’s Theorem. (CdS 10.1-11.3)
• Week 11: Dirac Structures: generalized tangent bundle and Courant bracket; examples: presymplectic forms, Poisson structures and integrable distributions; Lie algebroids and presymplectic foliation; Gauge transformations; admissible functions and Hamiltonian vector fields; Dirac maps and submanifolds; coisotropic and cosymplectic submanifolds of Poisson manifolds and Dirac theory of constrains. (Bursztyn’s Notes)
• Week 12: Generalized complex manifolds: complex structures on the generalized tangent bundle, complex Dirac structures, tensors associated with a GCS, gauge transformations and generalized holomorphic maps; type of a GCS and generalized Darboux theorem; generalized complex submanifolds; comments on Mirror Symmetry. (M. Gualtieri Ph.D Thesis)