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In the lecture notes, the author will survey the development of conformalgeometry on four dimensional manifolds. The topic she chooses is one on whichshe has been involved in the past twenty or more years: the study of theintegral conformal invariants on 4-manifolds and geometric applications. Thedevelopment was heavily influenced by many earlier pioneer works; recentprogress in conformal geometry has also been made in many different directions,here we will only present some slices of the development.

Conformal Geometry on Four Manifolds