Introduction
1Introduction to Noncommutative Geometry
1.1Topology and C*-algebras
1.1.1Definitions
1.1.2Spectral theory
1.1.3Duality in tile commutative case
1.1.4GNS construction
1.1.5Vector bundles and projective modules
1.2Measure theory and yon Neumann algebras
1.2.1Definition of von Neumann algebras
1.2.2Duality in the commutative case
1.3Noncommutative differential geometry
1.3.1Algebraic geometry
1.3.2Differential calculi
1.3.3Hochschild and cyclic homologies
1.3.4Spectral triples
2Epsilon-graded algebras noncommutative geometry
2.1General theory of the ε-graded algebras
2.1.1Commutation factors and multipliers
2.1.2Definition of ε-graded algebras and properties
2.1.3Relationship with superalgebras
2.2Noncommutative ε-graded geometry
2.2.1Differential calculus
2.2.2ε-connections and gauge transformations
2.2.3Involutions
2.3Application to some examples of ε-graded algebras
2.3.1ε-graded commutative algebras
2.3.2ε-graded matrix algebras with elementary grading
2.3.3ε-graded matrix algebras with fine grading
3An Introduction to Renormalization of QFT
3.1Renormalization of scalar theories in the wilsonian approach
3.1.1Scalar field theory
3.1.2Effective action and equation of the renormalization grour
3.1.3Renormalization of the usual 4 theory in four dimensions
3.2BPHZ renormalization
3.2.1Power-counting
3.2.2BPHZ subtraction scheme
3.2.3Beta functions
3.3Renormalization of gauge theories
3.3.1Classical theory and BRS formalism
3.3.2Algebraic renormalization
4QFT on Moyal space
4.1Presentation of the Moyal space
4.1.1Deformation quantization
4.1.2The Moyal product on Schwartz functions
4.1.3The matrix basis
4.1.4The Moyal algebra
4.1.5The symplectic Fourier transformation
4.2UV/IR m/x.ing on the Moyal space
4.3Renormalizable QFT on Moyal space
4.3.1Renormalization of the theory with harmonic term
4.3.2Principal properties
4.3.3Vacuum configurations
4.3.4Possible spontaneous symmetry breaking?
4.3.5Other renormalizable QFT on Moyal space
5Gauge theory on the Moyal space
5.1Definition of gauge theory
5.1.1Gauge theory associated to standard differential calculus
5.1.2U(N) versus U(1) gauge theory
5.1.3UV/IR mixing in gauge theory
5.2The effective action
5.2.1Minimal coupling
5.2.2Computation of the effective action
5.2.3Discussion on the effective action
5.3Properties of the effective action
5.3.1Symmetries of vacuum configurations
5.3.2Equation of motion
5.3.3Solutions of the equation of motion
5.3.4Minima of the action
5.3.5Extension in higher dimensions
5.4Interpretation of the effective action
5.4.1A superalgebra constructed from Moyal space
5.4.2Differential calculus and scalar theory
5.4.3Graded connections and gauge theory
5.4.4Discussion and interpretation
Conclusion
Bibliography
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