Title:  Deformations of curves and surfaces in a variety.

Abstract:  I will briefly review the deformation theory of the Hilbert 
scheme and the flag Hilbert scheme.  Then I will discuss the problem of 
deforming reducible, genus 0 curves in a variety.  Here the key notions 
are "free rational curves" and "very free rational curves".  I will use 
this to construct the quotient by free rational curves and to prove an 
important criterion for rational connectedness -- existence of a very 
free curve.  Time permitting, I will discuss the problem of deforming 
unions of rational surfaces in a variety.  Here the key notions are 
"twisting surfaces" and "very twisting surfaces".