Mathematics Colloquium

Riemann predicted in his inaugural dissertation that the microstructure of space may not be a manifold because we do not know how light and other bodies will behave in the ultra-small regions of space. The physicists have invented a new geometry called supergeometry, in which the local rings of spacetime are generated by both commuting and anti-commuting elements. Supermanifolds make sense and their automorphisms are super Lie groups. Unitary representations of super Lie groups make sense. The classification of irreducible unitary representations of the super Poincare group gives the superparticles. The theory predicts that each superparticle consists of a set of paired ordinary particles of opposite spin, and each member of a pair is called a partner of the other. The discovery of the superpartners is the main goal of the CERN collider. If confirmed, these superpartners will affirm that supersymmetry is a symmetry in Nature, not merely in mathematics. Unfortunately, the breaking of supersymmetry long ago forced the superparners to have much higher mass and so it takes very high energies to find them. It is a delicate situation because the energy levels of supersymmetry breaking are more or less the same as the energy levels which are reachable in the CERN collider.