Pasha Belorousski,
Introduction to Moduli Problems in Algebraic Geometry.
Moduli problems are ubiquitous in algebraic geometry. Solutions
to moduli problems are called "moduli spaces". Every scheme, for
example, is the moduli space for a certain (even if not
particularly interesting) moduli problem associated to it. This
is the point of view due to Grothendieck - considering schemes as
"functors of points". Moduli spaces for more interesting problems
include the projective space, Grassmannians, flag varieties,
Hilbert schemes, Jacobians, Picard varieties, moduli spaces of
curves, moduli spaces of vector bundles, etc. etc.
In this talk I will describe the functorial point of view on
algebraic geometry, go over basic definitions in moduli theory
and discuss some of the examples mentioned above. I hope to give
one or two followup talks later in the semester about moduli
spaces of curves, which in my view constitute the richest
examples in moduli theory.