Read e-book online An invitation to noncommutative geometry PDF

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 present quantity that collects lectures in this vital and speedy constructing topic in arithmetic. The lectures are given through prime specialists within the box and the variety of issues is stored as wide as attainable through together with either the algebraic and the differential features of noncommutative geometry in addition to fresh functions to theoretical physics and quantity concept.

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

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13) as C ∞ (T2θ ), where we think of T2θ as the (smooth) noncommutative torus. 5) it is not necessary to restrict to the condition that i, j lie on the transversal T . 14) where now the parameter of integration is no longer discrete. This ought to correspond to the same noncommutative space, and in fact the algebras are related by Bθ = Aθ ⊗ K , where K is the algebra of all compact operators. The tangent space to the ordinary torus T 2 is spanned by the tangent ∂ ∂ directions ∂x and ∂y . By choosing coordinates U, V , with U = e2πix ∂ ∂ and V = e2πiy , the tangent vectors are given by ∂x = 2πiU ∂U and ∂ ∂ ∂y = 2πiV ∂V .

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 .

196] and references therein). In the presence of a natural smooth subalgebra of Cr∗ (Γ) containing the group ring and stable under holomorphic functional calculus, the combination of the two steps described above makes it possible to prove an index theorem which is an higher dimensional form of Atiyah’s L2 -index theorem for coverings. This gave the first proof of the Novikov conjecture for hyperbolic groups ([83]). Since then the analysis of dense smooth subalgebras has played a key role, in particular in the ground-breaking work of Vincent Lafforgue.

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An invitation to noncommutative geometry by Masoud Khalkhali


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