Research on invariants of real analytic singularities

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The aim of this research is to look for and introduce some invariants of real analytic singularities on topological equivalence, blow-analytic equivalence, Lipschitz equivalence, C^1 equivalence and so on, and to give classifications of real analytic singularities on those equivalences using the invariants and triviality theorems. Concerning these problems, I have obtained the following :1)In the joint work with Adam Parusinski, we have introduced motivic-type invariants as blow-analytic invariants for real analytic function germs, and have given blow-analytic classification of real analytic functions using our invariants and the Fukui invariants.2)I have worked with A.Parusinski also on a different type invariant from the above for real analytic functions of 2 variables. We have shown that the Newton boundary relative to an arc is a Lipschitz invariant and the Newton boundary with dots relative to an arc is a C^1 invariant in the 2 variables case.3)The Fukui invariant is well-known as … More a blow-analytic invariant for real analytic functioin germs. On the other hand, there is a problem whether the Fukui invariant is a topological invariant for complex holomorphic function germs. I have given the affirmative answer to it in the 2 variables case, and have constructed a negative example of 4 variables functions.4)For a semialgebraic mapping between semialgebraic sets, I and Mahiro Shiota have studied the set of points at which the fibre is not smooth. In particular, we have looked for an invariant for the triviality along the smooth part of a fibre. Then we have proved that a semialgebraic function is semialgebraically trivial along the smooth fibre, and constructed an example of a semialgebraic mapping which is not trivial along it.5)Related to the problem on Blow-Nash moduli for a family of algebraic singularities, I have proved with T.Fukui and K.Bekka that any polynomial mapping can be represented as the restriction of a desingularisation map of some algebraic variety to the intersection of the strict transform of the variety and the exceptional set at some point.6)I have introduced the notion of a sea-tangle neighborhood for a Lipschitz arc, and have shown that the degree and the width of the neighborhood are biLipschitz invariants and that the Briancon-Speder family and the Oka family are not biLpischitz trivial using the invariants. In addition, developing the idea, I have obtained a general result with Laurentiu Paunescu that the kissing dimension of subanalytic sets is a biLipschitz invariant. Less