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For the true Ricci flow part of Thomas' program, you can use the new B3MP (Bessières, Besson, Boileau, Maillot, Porti) book available at http://www-fourier.ujf-grenoble.fr/~besson/english_principal.pdf and to appear at EMS. It is aimed at explaining Ricci flow with surgery (or rather a variation called Ricci flow with bubbling-off) and the proof of geometrization to topologists and geometers, and the analysis of Ricci flow is mostly used as a blackbox, so that may suit you or not.