Derived and modular resolutions of the Stable map moduli and applications

In this talk, I will present a derived version of the  resolutions of the moduli spaces of stable maps
(for all genera). This  resolution can be used to rigorously define the so-called reduced  Gromov-Witten
numbers of Calabi-Yau threefolds (i.e., the GW numbers  associated to the main components of the
stable map moduli).

The derived resolutions are singular (in the usual sense). For further  applications, a resolution
(in the usual sense) is desirable (for  example, Zinger's proof of genus-1 mirror formula uses such a 
resolution); I will describe how to achieve this by a sequence of  modular blowups in the case of
genera 1 and 2.  All are ioint work  with Jun Li.