Limits in Hodge Theory

Almost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the literature the proofs of these results frequently use the consequences of the analysis of the singularities acquired in a degenerating family of Hodge structures; that analysis itself is treated as a "black box." In these lectures an attempt will be made to give an informal introduction to the subject of limits of Hodge structures and to explain some of the essential ideas of the proofs. One additional topic not yet in the literature that we will discuss is the geometric interpretation of the extension data in limiting mixed Hodge structures and its use in moduli questions.