Eduardo Esteves: Limit linear series: History and Perspectives.

The theory of limit linear series was created by Eisenbud and Harris in the 80's and has been ever so useful since then, especially in describing divisors in moduli spaces of curves. Unfortunately, the theory is set up only for curves of compact type. Since then, some attempts were made to generalize the theory. However, as time passed by, it became clear that a new framework for curves of compact type had to be considered. Osserman gave a new definition of limit linear series in 2006, that proved to be partially useful in describing  limits of divisors. I will review the history of the theory, explaining the motivation and the fundamentals, and then present a new notion of limit linear series, one that describes completely limits of divisors. This is joint work with Antonio Nigro (Niterói) and Pedro Rizzo (Medellín).