https://sites.google.com/view/gabrielebenedetti/teaching
All Monday lectures are in room A136. All Wednesday lectures are in room A133.
This course introduces the core ideas of symplectic geometry, from the foundations to the current frontier of research. Symplectic geometry yields a powerful mathematical description of physical phenomena from the classical to the quantum level. Symplectic manifolds model phase spaces of systems, their Lagrangian submanifolds representing natural boundary conditions for the dynamics, which is encoded in a Hamiltonian function. Hamiltonian functions play a crucial dual role since they describe both the dynamics and the symmetries in symplectic geometry, as stated by Noether's principle. Beside building up the fundamentals of symplectic geometry, the lectures will have the following three learning goals.
GOAL 1: Construct and obstruct embeddings of one symplectic manifold into another. In particular, we will prove Gromov's Non-Squeezing Theorem.
GOAL 2: Classify Lagrangian submanifolds, up to symmetry, and study their (self)intersections. In particular, we will show that there is no exact Lagrangian in euclidean space, and discuss Arnold's conjecture.
GOAL 3: Study the group of symmetries of symplectic manifolds. In particular, we will define Hofer's metric, and investigate the relationship between symplectomorphisms and (volume-preserving) diffeomorphisms.
To achieve our goals, we will learn to use pseudoholomorphic curves, whose theory was initiated by Gromov and Floer in the 1980s.
Prerequisites
We will only assume basic notions from Differential Geometry and Topology. No prior exposure to symplectic geometry is needed. The course is suitable also to interested Master students.
Literature
Our main sources for the course are
i) Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2001.
ii) McDuff and Salamon, Introduction to Symplectic Topology, Third Edition, Oxford University Press, 2016.
iii) McDuff and Salamon, J-Holomorphic Curves and Symplectic Topology, Second Edition, AMS, 2012.
iv) Brendel, An Introduction to the Chekanov Torus, Proceedings of the 27th Goköva geometry-topology conference 2021, 2022.
Exam
The exam will consist in a 50-minute seminar talk with questions. We will choose together the topic of the talk according to your interests and background. We expect that students attend the seminar talks of the other participants.