Characteristic p method in singularity theory

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We applied "characteristic p" methods to singularity theory and obtained the fokkowing results.1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q- Gorenstein. The same is true for rational singularities and F-rational rings. Also, we found that the same is true for "singularity of pairs" (KLT, PLT etc.). The latter result is a joint work with N.Hara.2 The notion of "Hilbert-Kunz multiplicity", which is a new "multiplicity" for local rings. We characterized regular local rings, certain rational singularities in dimension 2 by this multiplicity. Also we found a very beautiful and mysterious formula for integrally closed ideals in 2-dimensional rational double points. (a joint work with K.Yoshida)3 We investigated chains of integrally closed ideals and found the existence of a composition series only by integrally closed ideals. We found that the family of integrally closed ideals of colength 1 corresponds to the closed points of the fiber cone. Also, we found a new characterization of simple integrally closed ideals in 2-dimensional regular local rings. (The last result is a joint work with S.Noh.)