By Masoud Khalkhali

This can be the 1st present quantity that collects lectures in this vital and speedy constructing topic in arithmetic. The lectures are given by means of top specialists within the box and the diversity of themes is saved as wide as attainable by way of together with either the algebraic and the differential facets of noncommutative geometry in addition to contemporary purposes to theoretical physics and quantity concept.

**Contents: **

- A stroll within the Noncommutative backyard (A Connes & M Marcolli);
- Renormalization of Noncommutative Quantum box idea (H Grosse & R Wulkenhaar);
- Lectures on Noncommutative Geometry (M Khalkhali);
- Noncommutative Bundles and Instantons in Tehran (G Landi & W D van Suijlekom);
- Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori (S Mahanta);
- Lectures on Derived and Triangulated different types (B Noohi);
- Examples of Noncommutative Manifolds: advanced Tori and round Manifolds (J Plazas);
- D-Branes in Noncommutative box thought (R J Szabo).

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**Additional resources for An invitation to noncommutative geometry**

**Example text**

A Walk in the Noncommutative Garden 29 The classical Hall eﬀect is a physical phenomenon ﬁrst observed in the 19th century [121]. A very thin metal sample is immersed in a constant uniform strong magnetic ﬁeld orthogonal to the surface of the sample. By forcing a constant current to ﬂow through the sample, the ﬂow of charge carriers in the metal is subject to a Lorentz force perpendicular to the current and the magnetic ﬁeld. 1) where E is the electric ﬁeld, e and N the charge and number of the charge carriers in the metal, B the magnetic ﬁeld, and j the current.

Another very signiﬁcant noncommutative example is the Standard Model of elementary particles (cf. [63]), which we discuss in Section 17. Another example of spectral triple associated to a classical space, which is not classically a smooth manifold, is the case of manifolds with singularities. In particular, one can consider the case of an isolated conical singularity. This case was studied by Lescure [148]. Let X be a manifold with an isolated conical singularity. The cone point c ∈ X has the property that there is a neighborhood U of c in X such that U {c} is of the form (0, 1] × N , with N a smooth compact manifold, and metric g|U = dr2 + r2 gN , where gN is the metric on N .

The integer values of the Hall conductance are observed with a surprising experimental accuracy of the order of 10−8 . This phenomenon of quantization of the Hall conductance is known as the Integer Quantum Hall Eﬀect (IQHE). Laughlin ﬁrst suggested that IQHE should be of a geometric origin [145]. A detailed mathematical model of the IQHE, which accounts for all the important features of the experiment (quantization, localization, insensitivity to the presence of disorder, vanishing of direct conductance at plateaux levels) improving over the earlier Laughlin model, was developed by Bellissard and collaborators [18], [19].