Spectral action in noncommutative geometry / Michał Eckstein, Bruno Iochum.

Intro; Preface; References; Contents; 1 The Dwelling of the Spectral Action; 1.1 Spectral Triples; 1.2 Some Spaces Associated with mathscrD; 1.3 Abstract Pseudodifferential Calculus; 1.4 Dimension Spectrum; 1.5 Noncommutative Integral; 1.6 Fluctuations of Geometry; 1.7 Intermezzo: Quasi-regular Spectral Triples; 1.8 The Spectral Action Principle; References; 2 The Toolkit for Computations; 2.1 Spectral Functions; 2.2 Functional Transforms; 2.2.1 Mellin Transform; 2.2.2 Laplace Transform; 2.3 Weyl's Law and the Spectral Growth Asymptotics; 2.4 Poisson Summation Formula

2.5 Asymptotic Expansions2.6 Convergent Expansions; 2.7 On the Asymptotics of Distributions; References; 3 Analytic Properties of Spectral Functions; 3.1 From Heat Traces to Zeta Functions; 3.2 From Zeta Functions to Heat Traces; 3.2.1 Finite Number of Poles in Vertical Strips; 3.2.2 Infinite Number of Poles in Vertical Strips; 3.3 Convergent Expansions of Heat Traces; 3.3.1 Convergent, Non-exact, Expansions; 3.3.2 Exact Expansions; 3.4 General Asymptotic Expansions; 3.5 Asymptotic Expansion of the Spectral Action; References; 4 Fluctuations of the Spectral Action

4.1 Fluctuations of the spectral zeta-function4.2 Fluctuations of Noncommutative Integrals; 4.3 Consequences for the Spectral Action; 4.4 Operator Perturbations; References; 5 Open Problems; References; A Classical Tools from Geometry and Analysis; A.1 About ``Heat Operators''; A.2 Definition of pdos, Sobolev Spaces and a Few Spectral Properties; A.3 Complex Parameter-Dependent Symbols and Parametrix; A.4 About e-tP as a pdo and About its Kernel; A.5 The Small-t Asymptotics of e-tP; A.6 Meromorphic Extensions of Certain Series and their Residues; B Examples of Spectral Triples; B.1 Spheres; B.2 Tori; B.3 Noncommutative Tori; B.3.1 Dimension Spectrum; B.3.2 Heat Kernel Expansion; B.3.3 Spectral Action for Noncommutative Tori; B.4 Podleś Sphere; References; Index