Derived intersections

In recent years there have been remarkable developments in derived algebraic geometry, a field which lies at the confluence of algebraic geometry and algebraic topology. The main objects of study in derived algebraic geometry are dg schemes, and a typical example of dg schemes arises when studying intersection theory of subvarieties. In my talk I shall try to explain some of the fundamental ideas of derived algebraic geometry, and argue that it allows us to get a more geometric understanding of classical results like the proof by Deligne-Illusie of the Hodge-de Rham degeneration.