Reading Group in Symplectic Geometry
Fall Quarter 2018
University of California Davis
Department of Mathematics
This is the website for the Reading Group in Symplectic Geometry at UC Davis.
We meet regularly on Wednesday at 11:00am in Room MSB 3240, see the schedule for topics below.
The course will start with us going through Ana Cannas da Silva's book "Lectures on Symplectic Geometry".
The second part of the course will be topics. The picture on the right depicts Lagrangian caustics in the wild.
For more Lagrangian singularities, check this project at the Laser Teaching Center in Stony Brook University.
Participants: Jennifer Brown, Cooper Jacob, Black Fushuai Jiang, Matthew Lin, Beibei Liu, John Yoon Seok Chae, and Haihan Wu.
Contact Roger Casals, casals -at- math.ucdavis.edu, for questions and suggestions if you are interested.
- Tuesday Sep 11: The reading assignment is Chapter I with the exercises therein.
We discussed symplectic orthogonals in detail. We studied examples of isotropic, coisotropic, Lagrangian and symplectic linear subspaces.
Particular emphasis was given to the 4-dimensional case, and deciding whether a 2-plane is symplectic or Lagrangian. Briefly discussed the Lagrangian Grassmannian U(n)/O(n).
- Tuesday Sep 18: The reading assignment is Chapter II, with the exercises therein.
We learned in detail how to compute and draw the Liouville vector associated to a Liouville form.
In particular we explored three different Liouville vector fields in the cotangent bundle of R. Discussion on cohomological obstructions to exactness (Stokes).
- Wednesday Sep 26: The reading assignment is Chapter III, with the exercises therein.
We discussed examples of Lagrangian submanifolds as graphs of 1-forms. Emphasis on the graph of df where f is a function.
We elaborated on the construction of symplectomorphisms as flows of Liouville fields. Comparison between volume-preserving and symplectomorphism.
Small list of examples where the symplectomorphism group is known. Statement of Gromov non-squeezing theorem.
- Wednesday Oct 3: The reading assignment is Chapter IV, with the exercises therein.
The topic of the meeting was generating functions. We ended up focusing particularly on Morse Theory.
Basics of Morse Theory, trajectories and the Morse complex. Infinite dimensional Morse Theory.
Examples of Functionals. Yang-Mills, Chern-Simons and the Floer Action Functionals. Instantons.
- Wednesday Oct 10: The reading assignment is Chapter VI, with the exercises therein.
The discussion centered around smooth tubular neighborhoods of smooth submanifolds and the theory of vector bundles.
This included examples of smooth submanifolds with non-trivial normal bundles. Real Projectives Lines and Conics in the Real Projective Plane.
Complex curves in the Complex Projective Plane. Rank two vector bundles over the 2-sphere and Euler class.
- Wednesday Oct 17: The reading assignment are Chapters VII and VIII, with the exercises therein.
We discussed symplectic neighborhoods of isotropic and Lagrangian submanifolds.
The main application is the topological restriction, e.g. the only orientable Lagrangians in 4-space are tori.
Short discussion of Lagrangians in projective space (as the real locus) and diagonals.
- Wednesday Oct 24: The reading assignment are Chapters X, with the exercises therein.
Discussion on distributions, foliations and the Frobenius integrability theorem.
Examples of integrability for real distributions, almost complex structures and Kahler metrics.
Maximal non-integrability. Integral submanifolds and Legendrian submanifolds. Connection to Lagrangians.
- Wednesday Oct 31: "Symplectic capacities and the uncertainty principle" by B. Jiang.
The main reference is Symplectic Methods in Harmonic Analysis and in Mathematical Physics by M. de Gosson.
- Wednesday Nov 7: "Morse theory for Lagrangian intersections" by Y. Seok Chae
The main reference is Morse theory for Lagrangian intersections by A. Floer,
and also Lectures in Floer Homology by D. Salamon.
- Wednesday Nov 14: "On the Goldberg Conjecture" by C. Jacob
The main references are On some compact Einstein almost Kaehler manifolds by K. Sekigawa,
and also On some 4-dimensional almost Kaehler manifolds by Tedi C. Draghici.
- Wednesday Nov 28: "D-modules and Symplectic Geometry" by J. Brown
The main references are A Primer of Algebraic D-Modules by S. C. Coutinho, and also D-Modules, Perverse Sheaves, and Representation Theory by R. Hotta, K. Takeuchi, and T. Tanisaki.
In addition, discussion on symplectic actions by Lie groups, following Chapters XVIII, XXI, and XXII. Hamiltonian actions and vector fields. One-parameter groups of symplectomorphisms, circle actions. Converting between moment map and co-moment map perspective. Physical interpretation of the fibers of a moment map.