An invitation to noncommutative geometry by Masoud Khalkhali

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By Masoud Khalkhali

This can be the 1st current quantity that collects lectures in this very important and quick constructing topic in arithmetic. The lectures are given by means of prime specialists within the box and the variety of subject matters is stored as vast as attainable by way of together with either the algebraic and the differential features of noncommutative geometry in addition to contemporary functions to theoretical physics and quantity thought.


  • 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: complicated Tori and round Manifolds (J Plazas);
  • D-Branes in Noncommutative box thought (R J Szabo).

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Let M be a smooth compact Riemannian Spin manifold, and (A, H, D) be the corresponding spectral triple given by the algebra of smooth functions, the space of spinors, and the Dirac operator. Then the metric dimension agrees with the usual dimension n of M . The dimension spectrum of M is the set {0, 1, . . , n}, where n = dim M , and it is simple. ) A Walk in the Noncommutative Garden 51 It is interesting in the case of an ordinary Riemannian manifold M to see the meaning of the points in the dimension spectrum that are smaller than n = dim M .

Recall that the Brillouin zones of a crystals are fundamental domains for the reciprocal lattice Γ obtained via the following inductive procedure. The Bragg hyperplanes of a crystal are the hyperplanes along which a pattern of diffraction of maximal intensity is observed when a beam of radiation (Xrays for instance) is shone at the crystal. The N -th Brillouin zone consists of all the points in (the dual) Rd such that the line from that point to the origin crosses exactly (n − 1) Bragg hyperplanes of the crystal.

20]). This defines d A Walk in the Noncommutative Garden 37 Figure 10. Quasiperiodic tilings and zellijs. a locally compact groupoid G(L, X). The C ∗ -algebras C ∗ (G(L, X)) and C(Ω) T Rd are Morita equivalent. In the case where L is a periodic arrangement of points with cocompact symmetry group Γ ⊂ Rd , the space Ω is an ordinary commutative space, which is topologically a torus Ω = Rd /Γ. The C ∗ -algebra A is in this case ˆ ⊗ K, where K is the algebra of compact operators and isomorphic to C(Γ) ˆ Γ is the Pontrjagin dual of the abelian group Γ ∼ = Zd , isomorphic to T d , d obtained by taking the dual of R modulo the reciprocal lattice Γ = {k ∈ Rd : k, γ ∈ 2πZ, ∀ γ ∈ Γ} .

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