Research on singularities of a variety

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We gave an affirmative answer to the Nash problem for an arbitrary dimensional toric variety.We gave a negative answer to the Nash problem in general by showing a counter example of dimension 4.We studied the structure of the arc space of a toric variety and obtain that the dominating relation of the orbits is translated by the relation of valuations corresponding to the orbits.We gave an affirmative answer to the Nash problem for non-normal toric variety.We gave an affirmative answer to the local Nash problem for quasi-ordinary singularities.We defined a maximal divisorial set on the arc space of a variety and proved that any irreducible component of a contact locus is a maximal divisorial set.We proved that a maximal divisorial set is represented by the intersection of finite number of contact loci of functions.We proved that every integrally closed ideal of 2-dimensional regular local ring is a multiplier ideal.We gave a characterization of 2-dimensional Gorenstein singularities with arithmetic genus 1.