Dmitri Vassilevich: Spectral triples and heat kernel expansion

Main notions of noncommutative geometry: noncommutative topology and the Gelfand-Naimark Theorem, spectral triples and Dirac operators, noncommutative distance, spectral action.

Spectral geometry: differential and pseudodifferential operators, spectrum, spectral functions, heat kernel, asymptotic expansions.

Examples and applications: spectral asymptotics on some noncommutative spaces, applications to the Standard Model, high and low-energy behavior of the spectral action.

Literature

1. Basic NCG
   a. J. Gracia-Bondia, J. Varilly and H. Figueroa, "Elements of noncommutative geometry", Birkhauser, 2001 [about 600 pages!]
   b. J. Varilly, "An introduction to noncommutative geometry", EMS, 2006 [just about 100 pages]
   
2. Spectral geometry (physics-friendly)
   a. D. Vassilevich, "Heat kernel expansion: users manual", Phys. Rep. 388 (2003) 279-360 [arXiv:hep-th/0306138]
   b. D. Fursaev and D. Vassilevich, "Operators, Geometry and Quanta: Methods of spectral geometry in quantum field theory", Springer, 2011.
   
3. NCG and particle physics
   a. A. Connes and M. Marcolli, "Noncommutative geometry, quantum fields and motives" AMS, 2008 [760 pages!]
   b. W. van Suijlekom, "Noncommutative geometry and particle physics", Springer, 2015 [a nice introduction]