Ma 140 b: Noncommutative Geometry, Part I

Winter 2010, Caltech Math Department Instructor: Matilde Marcolli

Brief Course Description

This will be a first introductory course on noncommutative geometry, which will include a quick review of the needed notions from operator algebra and a detailed discussion of the main tools such as cyclic cohomology, spectral triples, KMS states, and with a focus on examples of noncommutative spaces of relevance to physics.

Prerequisites:

140a is recommended, but a course in quantum mechanics also constitutes a valid prerequisite: both mathematics and physics students are encouraged to attend.

Syllabus:

C* algebras, Gelfand-Naimark, topological noncommutative spaces, crossed product C* algebras and noncommutative quotients, K-theory of C* algebras, Hilbert modules, Fredholm modules and KK-theory, strong Morita equivalence.
von Neumann algebras, an outline of the theory of factors, quantum statistical mechanics and KMS states, Connes-Takesaki duality.

Moduli, noncommutative tori and elliptic curves, the noncommutative geometry at the boundary of modular curves, real multiplication.

The Connes bicomplex, Chern character, index theorems, cyclic modules and the cyclic category.
The Connes-Chern character and noncommutative geometry models of the quantum Hall effect, quasicrystals and aperiodic solids.

Spectral triples, dimension spectrum, finite and theta summability, zeta functions, the spectral action functional, asymptotic expansion.
Examples of spectral triples: manifolds, fractals, quantum groups, noncommutative tori, particle physics models, cosmological applications, spectral triples in quantum gravity.

Noncommutative spaces and dynamical systems, with applications to arithmetic geometry.