ABSTRACTS OF THE BLUMENTHAL LECTURES, 2012

Lagrangian submanifolds appear naturally in symplectic

geometry from different angles: real algebraic geometry, dynamics and

integrable systems are just a few sources to mention. In fact, once

the concept is introduced many symplectic phenomena can be rephrased

in a Lagrangian language, as a famous citation of A. Weinstein goes

"Everything is Lagrangian".

I will survey the development of Lagrangian topology, which is a

theory that mixes topological and symplectic invariants associated to

Lagrangian submanifolds. I will explain how these can be applied to

solve various problems in symplectic geometry. I will then move on to

newer developments that have to do with the algebraic structure of the

totality of these invariants, such as the Donaldson and Fukaya

categories. If time permits I will outline a new and more geometric

approach to study these categories.

No apriori knowledge of symplectic geometry is needed.