Symplectic geometry was built up under classical mechanics. Indeed, the phase space is the total space of the cotangent bundle to the configuration space, while any cotangent bundle always admits canonical symplectic structure. Another example class - of compact symplectic manifolds - comes from the considerations of complex submanifolds of complex projective spaces. These submanifolds are the central objects of algebraic geometry; at the same time they could be studied from the point of view of symplectic geometry. The main topic of the present lecture course is the properties of lagrangian submanifolds, related to the classification problem of lagrangian submanifolds. This subject appears in the case of integrable classical mechanical systems: the classical Liouville theorem says that the common level set for first integrals of a completely integrable system is a lagrangian submanifold (in the compact case - lagrangian torus). We discuss examples of lagrangian submanifolds, present the theory of local deformation of these submanifolds and introduce certain invariants of these submanifolds.

The course is intended for the students with primary knowledge of differential geometry. Since the course itself is short we will be focused mostly on geometric interpretation while omitting some particular details of the proofs.

Lecturer
Tyurin Nikolai Andreevich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center