Slides of my series of lectures on “Introduction to Noncommutative Geometry” at UC Berkeley in Spring 2015. (Many thanks to Prof. Marc Rieffel for letting me give these lectures in his Quantum Geometry seminar.)

Part 4: Index Theory and the Connes-Chern Character (pdf).

Slides of my short courses on noncommutative geometry at Yau Mathematical Science Center, Tsinghua University, Beijing, China (October 24–27, 2011) and Australian National University, Canberra, Australia (August 2012).

Notes of my graduate courses on noncommutative geometry at the University of Tokyo (Oct. 2010 – Jan. 2011) and Seoul National University (Spring 2012 and Spring 2018).

Chap. 8.: Clifford algebras, spinors, Dirac operators and the Atiyah-Singer index theorem (pdf).

Chap. 9-12: K-theory, cyclic cohomology, the local index formula in NCG, the CM cocycle for Dirac spectral triples, Connes-Moscovici’s index theorem in diffeomorphism-invariant geometry (pdf).