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Introduction to Noncommutative Geometry

Course Type: 
PhD Course
Academic Year: 
40 h

 These lectures focus on the latest ’layer’ Riemannian and Spin of Noncommutative Geometry (NCG). Its central concept, due to A. Connes, is ’spectral triple’ which consists of an algebra of operators on a Hilbert space and an analogue of the Dirac operator. A prototype is the canonical spectral triple of a Riemannian spin manifold which will be described starting with the basic notions of multi-linear algebra and differential geometry. Then few of its additional properties will be presented, which permit the reconstruction of the underlying geometry and enjoy fascinating generalizations to ’quantum’ spaces. Some earlier `layers' of NCG will be also briefly presented when needed, regarding the (differential) topology and calculus: the equivalence between (locally compact) topological spaces and commutative C*-algebras, and between vector bundles and finite projective modules, elements of K-theory, Hochschild and cyclic cohomology, the noncommutative integral, and others. Some of these were important motivations for, or even remain pillars of, the present-day NCG, which is still being actively constructed. A few fruitful constructions to be discussed are products, fluctuations (perturbations) by gauge potentials, and conformal rescalings of spectral triples. Besides another classical example of Hodge-de Rham spectral triple, the quantum examples to be described will be the noncommutative torus, quantum spheres and the almost-commutative geometry behind the Standard Model of fundamental particles. In the final part (if time permits) the issue of symmetries (isometries, diffeomorphisms) will be discussed and generalized to Hopf algebras, quantum groups, noncommutative principal bundles and their associated vector bundles, and applied to equivariant spectral triples. There is a great wealth of available material, not all of which can be presented in these lectures. This regards some well-established topics like index theory, of which only a few indispensable facts from the theory of the (elliptic) Laplace operator will be used. Such selectivity will hopefully lead us more directly to some of the active and interesting fields of current research. The presentation style should comply with "mathematical physics/physical mathematics". The prerequisites are basics of multilinear algebra, differential geometry and Hilbert space operators. 

Tuesdays and Thursdays 11:00 - 13:00
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