By Masoud Khalkhali

ISBN-10: 981270616X

ISBN-13: 9789812706164

ISBN-10: 9812707794

ISBN-13: 9789812707796

ISBN-10: 9812814337

ISBN-13: 9789812814333

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

**Contents: **

- A stroll within the Noncommutative backyard (A Connes & M Marcolli);
- Renormalization of Noncommutative Quantum box conception (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 concept (R J Szabo).

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

**Example text**

An odd Fredholm module over A is given by: (1) a representation of A in a Hilbert space H. (2) an operator F = F ∗ , F 2 = 1, on H such that [F, a] is a compact operator for any a ∈ A . An even Fredholm module is given by an odd Fredholm module (H, F ) together with a Z/2 grading γ, γ = γ ∗ , γ 2 = 1 of the Hilbert space H satisfying: (1) γa = aγ, for all a ∈ A (2) γF = −F γ. This deﬁnition is derived from Atiyah’s deﬁnition [12] of abstract elliptic operators, and agrees with Kasparov’s deﬁnition [130] for the cycles in Khomology, KK(A, C), when A is a C ∗ -algebra.

The noncommutative tori and the components of the Yang–Mills connections appear in the classiﬁcation of the BPS states in M-theory [71]. 6) where the Ui are unitary gauge transformations. The multiplicative commutator U1 U2 U1−1 U2−1 is then central and in the irreducible case its scalar value λ = exp 2πiθ brings in the algebra of coordinates on the noncommutative torus. The Xj are then the components of the Yang–Mills connections. The same picture emerged from the other information one has about M-theory concerning its relation with 11-dimensional supergravity and that string theory dualities can then be interpreted using Morita equivalence, relating the values of θ on an orbit of SL2 (Z).

1) such that ∇j (ξb) = (∇j ξ)b + ξδj (b) ∀ ξ ∈ S , b ∈ Aθ . 2) One checks that, as in the usual case, the trace of the curvature Ω = ∇1 ∇2 − ∇2 ∇1 , is independent of the choice of the connection. We can make the following choice for the connection: 2πis (∇1 ξ)(s) = − ξ(s) , (∇2 ξ)(s) = ξ (s) . 3) A Walk in the Noncommutative Garden 43 Notice that, up to the correct powers of 2πi, the total curvature of S is an integer. In fact, the curvature Ω is constant, equal to θ1 , so that the irrational number θ disappears in the total curvature, θ× 1θ .

### An invitation to noncommutative geometry by Masoud Khalkhali

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Categories: Algebraic Geometry