Twisted K-theory has its main origin in the paper with P. Donovan

http://www.numdam.org/item?id=PMIHES_1970__38__5_0

The objective of this lecture is to revisit the subject with more modern methods in

the light of new developments inspired by Mathematical Physics. See Witten (Arxiv

hep-th/9810188), J. Rosenberg

http://anziamj.austms.org.au/JAMSA/V47/Part3/Rosenberg.html

, Laurent-Gentoux, Tu, Xu (ArXiv math/0306138) and Atiyah, Segal (ArXiv math/0407054),

among many other authors.

The unifiyng theme in our presentation is the notion of K-theory of Z/2-graded algebras, from which most of the classical theorems in twisted K-theory are derived.

We also prove some new results in the subject : a Thom isomorphism using Fredholm

operators in an Hilbert space, explicit computations in the equivariant case and new

Adams operations.