Mathematics

 

I will 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 of multi-linear algebra and differential geometry. Then 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 exposed when needed, as regards the (differential) topology and calculus: the equivalence between (locally compact) 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 more. They served as motivations for, and still remain pillars of the present-day NCG, which is still being actively constructed.

 

A few fruitful constructions with spectral triples to be discussed are their products, fluctuations (perturbations) by gauge potentials, and conformal rescalings. Besides another classical} example of Hodge-de Rham spectral triple, a few quantum examples will be described: the noncommutative torus, quantum spheres and the almost-commutative geometry behind the Standard Model of fundamental particles.

 

In the final part the symmetries will be discussed, in particular isometries and diffeomorphisms. The former will be extended to Hopf algebras, quantum groups, noncommutative principal and associated vector bundles, and applied to equivariant spectral triples.

 

There is a great wealth of available material, which can be only glimpsed. 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 a 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 essentially basics of multilinear algebra, differential geometry and Hilbert space operators.