1 Preface.- 2 Notation Index I The Basics: 3 Geometrical Background.- 4 Quantum fields II Spectral geometry: 5 Operators and their spectra.- 6 Spectral functions.- 7 Non-linear spectral problems.- 8 Anomalies and Index Theorem III Applications: 9 Effective action.- 10 Anomalies in quantum field theories.- 11 Vacuum energy.- 12 Open strings and Born-Infeld action.- 13 Noncommutative geometry and field theory IV Problem solving: 14 Solutions to exercises.

The authors D. Vassilevich and D. Fursaev are acknowledged experts in spectral geometry and quantum gravity. D. Fursaev has published more than 60 articles in high profile science journals, and he was appointed the rector of Dubna International University in 2008. D. Vassilevich is a professor for mathematical physics at the Universidade Federal do ABC, Santo Andre (Brazil). He published more than 100 articles in scientific journals and is author of the book ""Fundamental Interactions - A Memorial Volume for Wolfgang Kummer" (2009).