The standard model and beyond in noncommutative geometry

Noncommutative geometry and the formulation of the standard model within it is reviewed. The phrasing within noncommutative geometry of a model of particle physics based on S(U(2) × U(3)) is attempted and found to be incompatible with the mathematical structure. Noncommutative geometry versions of unified theories based on SU(15) and SU(16) are found not to yield the necessary spontaneous symmetry breaking. An extension of the standard model which includes right-handed neutrinos (and no additional fermions) is shown to be compatible with Poincaré duality only if the number of right-handed neutrinos is not equal to three.