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Algebraic integral geometry

This is a revised version of the notes from the week-long course I gave at the Centre de Recerca Matematica, Barcelona, in September of 2010. The aim is to give a working overview of recent methods and results in "Blaschkean integral geometry" (i.e. the subject revolving around the kinematic formulas of Blaschke) in the wake of the revolutionary new methods introduced over the past ten years or so by S. Alesker. The most striking new feature is the extent to which the subject has been invaded by algebra.