An analytic approach to Briançon-Skoda type theorems

The Briançon-Skoda theorem can be seen as an effective version of the Hilbert Nullstellensatz and gives a connection between size conditions on holomorphic functions and ideal membership. The size conditions are captured algebraically by the notion of integral closure of ideals. Many techniques have been applied to prove the Briançon-Skoda theorem and variations of it. The first proof by Briançon and Skoda used L-theory. Later, Lipman and Tessier observed that residue calculus could be used to obtain an alternative proof, and inspired by this approach they generalized the theorem to an algebraic setting. Berenstein-Yger et al. developed further this residue method by introducing a division formula by Berndtsson into the picture. The theory of tight closure, introduced by Hochster and Huneke, was motivated by, and has been used to prove, the Briançon-Skoda theorem. This thesis explores how one can use analytic methods, including residue theory, to obtain Briançon-Skoda type theorems on singular varieties.