Preface.- Introductory Remarks -- The Non-Commutative Moyal-Weyl Spaces Rd -- The Fuzzy Sphere -- Quantum Non-Commutative Phi-Four -- The Multitrace Approach.- Non-Commutative Gauge Theory -- Appendix A - The Landau States -- Appendix B - The Traces TrtAtB and TrtAtBtCtD -- Index.

These lecture notes provide a systematic introduction to matrix models of quantum field theories with non-commutative and fuzzy geometries.  The book initially focuses on the matrix formulation of non-commutative and fuzzy spaces, followed by a description of the non-perturbative treatment of the corresponding field theories. As an example, the phase structure of non-commutative phi-four theory is treated in great detail, with a separate chapter on the multitrace approach. The last chapter offers a general introduction to non-commutative gauge theories, while two appendices round out the text. Primarily written as a self-study guide for postgraduate students – with the aim of pedagogically introducing them to key analytical and numerical tools, as well as useful physical models in applications – these lecture notes will also benefit experienced researchers by providing a reference guide to the fundamentals of non-commutative field theory with an emphasis on matrix models and fuzzy geometries.

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