Abstract: The Jacobian
variety of a smooth complex curve is a complex torus that
admits two different algebraic descriptions. The Jacobian
can be described as the Picard variety, which is the moduli
space of line bundles, or it can be described as the
Albanese variety, which is the universal abelian variety
that contains the curve. I will talk about how to extend a
family of Jacobians varieties by adding degenerate fibers.
Corresponding to the two different descriptions of the
Jacobian are two different extensions of the Jacobian: the
Néron model, constructed by Néron, and the
relative moduli space of stable sheaves, constructed by
Langer, Maruyama, Simpson, and others. I will explain what
these two extensions are and then prove that they are
equivalent. This equivalence has surprising consequences for
both the Néron model and the moduli space of stable
sheaves, which I will explain.