Hodge theory is the use of differential operators to study the cohomology groups of a smooth manifold. At first, we will talk about the de Rham complex of a Riemannian manifold and introduce the Laplacian operator and harmonic forms. We will sketch a proof of Hodge's theorem that there is a unique harmonic representative in a de Rham cohomology class.

We will then move on to the setting of compact complex manifolds and discuss how the de Rham and Hodge theory interact with the complex structure. We will introduce $(p,q)$-forms and the Dolbeault operators and talk about the the Hodge decomposition for a complex projective variety. From the Hodge decomposition, we get a list of numerical invariants which are organized into the 'Hodge diamond' - we will compute these invariants in some examples.

This talk should be accessible to all first year grad students, although knowledge of some algebraic geometry or differential geometry will be useful. Speaker(s): Sanal Shivaprasad (UM)