Abstract:

Deformation theory is a way of understanding how mathematical objects vary in small families. Here the small-ness if reflected by working with the smallest rings: Artin local rings. (Notice that Artin local rings are small objects in commutative ring theory, hence algebraic geometry; development of deformation theory in other situation would require first specifying the meaning of small.)

In this talk we will describe a very, but not the most, general setting of deformation theory in terms of functors on categories of Artin rings. We will explain the notions of representability, hull, deformation-obstruction theories. Applications and detailed examples will be covered in later talks.

Notes: This is the first talk of a series of three talks on deformation theory. It contains necessary background materials for the later two talks, so it is strongly recommended for those interested in the topics covered in the other two talks to attend this one. (These topics are deformation theory in algebraic geometry, and deformation of Galois representations.)