Past lectures

Abstract

This lecture looks back to Lefschetz' book about the topology of algebraic varieties (1924). Take an (affine) algebraic variety, defined by a system of (complex) polynomial equations. Lefschetz' approach goes like this. Intersect the variety with a hyperplane. Then, study what happens to this intersection when the hyperplane moves in a parallel family. This allows one to understand the topology of the original variety, in terms of one hyperplane section and some additional data on it. The process can be iterated, and in the end yields a combinatorial description of our variety. In particular, topological cycles (homology classes) can be studied in this way. At present, we're beginning to see that the same strategy can be used to understand the symplectic geometry (rather than just the topology) of algebraic varieties. Instead of homology classes, one looks at a more refined class of cycles, namely Lagrangian submanifolds. This will be the actual subject of the lecture. One should expect lots of pictures, and on the other hand, a quantity of pure algebra (of a similar kind to what one gets in algebraic topology classes).