Irregular Hodge theory: Applications to arithmetic and mirror symmetry

Irregular Hodge theory was initiated by Deligne in 1984, with arithmetic motivations, to deal with Hodge properties in the context of connections with irregular singularities. It also proves useful for computing Hodge numbers of some classical motives. The talk will explain the origin and motivations, the main examples in exponential Hodge theory and, if time permits, examples of irregular Hodge-Tate structures related to Fano toric manifolds.