Mathematics

The course focuses 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. 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.