Application of jet schemes to singularities in positive characteristic.

Recently "Mather-Jacobian discrepancy" is introduced, which is an
analogy of usual discrepancy.  We can classify singularities by means
of MJ-discrepancy, in the similar way as usual discrepancy.  One
advantage of MJ-discrepancy is that is is well described by jet
schemes in arbitrary characteristic (joint work with A. Reguera).  In
the talk I will pose some conjectures about jet schemes and show that
these yield basic properties of singularities of positive
characteristic.  And then I will show that in lower dimensional
singularities, the conjectures hold.