Dixmier traces and some applications in non-commutative geometry

This is a discussion of recent progress in the theory of singular traces on ideals of compact operators, with emphasis on Dixmier traces and their applications in non-commutative geometry. The starting point is the book Non-commutative geometry by Alain Connes, which contains several open problems and motivations for their solutions. A distinctive feature of the exposition is a treatment of operator ideals in general semifinite von Neumann algebras. Although many of the results presented here have already appeared in the literature, new and improved proofs are given in some cases. The reader is referred to the table of contents below for an overview of the topics considered.