Hodge theory is a central area of algebraic geometry, and one of its celebrated achievements is Deligne's Mixed Hodge structures introduced in the 70s. These structures generalize classical Hodge theory, extending it from the cohomology of smooth projective varieties to that of all varieties.

The goal of this talk is to introduce Mixed Hodge structures for smooth varieties and see some of their applications. In particular, we will prove the Global Invariant Cycle Theorem.

If time permits, we will also discuss an application to the cohomology of algebraic varieties and a connection between Hodge theory and singularities.