Overview

Noncommutative differential geometry is a novel approach to geometry, aimed in part at applications in physics. It was founded in the early eighties by the 1982 Fields Medalist Alain Connes on the basis of his fundamental works in operator algebras. It is now a very active branch of mathematics with actual and potential applications to a variety of domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field. It is an important topic both for mathematics and physics.Sample Chapter(s)Chapter 1: The Local Index Formula in Noncommutative Geometry Revisited (820 KB) Contents: K-Theory and D-Branes, Shonan: The Local Index Formula in Noncommutative Geometry Revisited (Alan L Carey, John Phillips, Adam Rennie and Fedor A Sukochev) Semi-Finite Noncommutative Geometry and Some Applications (Alan L Carey, John Phillips and Adam Rennie) Generalized Geometries in String Compactification Scenarios (Tetsuji Kimura) What Happen to Gauge Theories under Noncommutative Deformation? (Akifumi Sako) D-Branes and Bivariant K-Theory (Richard J Szabo) Two-Sided Bar Constructions for Partial Monoids and Applications to K -Homology Theory (Dai Tamaki) Twisting Segal's K -Homology Theory (Dai Tamaki) Spectrum of Non-Commutative Harmonic Oscillators and Residual Modular Forms (Kazufumi Kimoto and Masato Wakayama) Coarse Embeddings and Higher Index Problems for Expanders (Qin Wang) Deformation Quantization and Noncommutative Geometry, RIMS: Enriched Fell Bundles and Spaceoids (Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul) Weyl Character Formula in KK-Theory (Jonathan Block and Nigel Higson) Recent Advances in the Study of the Equivariant Brauer Group (Peter Bouwknegt, Alan Carey and Rishni Ratnam) Entire Cyclic Cohomology of Noncommutative Manifolds (Katsutoshi Kawashima) Geometry of Quantum Projective Spaces (Francesco D'Andrea and Giovanni Landi) On Yang-Mills Theory for Quantum Heisenberg Manifolds (Hyun Ho Lee) Dilatational Equivalence Classes and Novikov-Shubin Type Capacities of Groups, and Random Walks (Shin-ichi Oguni) Deformation Quantization of Gauge Theory in ℝ4 and U (1) Instanton Problems (Yoshiaki Maeda and Akifumi Sako) Dualities in Field Theories and the Role of K -Theory (Jonathan Rosenberg) Dualities in Field Theories and the Role of K -Theory (Jonathan Rosenberg) Readership: Researchers and graduate students in Mathematical Physics and Applied Mathematics.

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