Foto Amedeo Avogadro
Scienze MFN Alessandria
26-31 March 2007: Noncommutative Spacetime Geometries

The history of noncommutative geometry goes back to the days of quantum mechanics, where observables like position and momentum no longer commute. In a noncommutative space the coordinates, x,y,... do not commute, and a prominent example is indeed canonical noncommutativity xy -yx=ih. The topology and geometry of these spaces has gradually developed so that vector bundles, de Rham cohomology and characteristic classes are meaningful concepts also in the noncommutative context. A related more algebraic approach has led to quantum groups (noncommutative groups) and their homogeneous spaces. There are several applications in theoretical physics. Field and gauge theories on noncommutative spaces provide new and alternative descriptions of physical systems; the study of instantons is particularly rich. Noncommutativity is also natural in string theory with fluxes, and the role of noncommutative deformations of gravity is under intense investigations; there the Poincare' group is deformed into a quantum Poincare' group. More and more renormalizable noncommutative field theories in 4 dimensions are known. Noncommutative (internal) spaces are also used to build new Kaluza-Klein type particle theories (where the tower of Kaluza-Klein modes is now finite). Other areas of applications include integrable systems (where quantum groups originated from) and the quantum Hall effect.
The conference aim is to bring together and foster exchange of ideas between researchers in all of these areas, that frequently have different backgrounds. Toward this aim, and to benefit young researchers, a series of minicourses has been included. These minicourses will be devoted to the recent achievements in:
1) the structure of noncommutative symmetries of deformed spacetime (turning on noncommutativity requires a generalized notion of symmetry principles, that is as powerful as the classical one).
2) The standard model obtained from a ten dimensional theory where the six internal spacetime dimensions are noncommutative.
3) Noncommutative instantons and their moduli spaces.
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