Research on invariants of real analytic singularities

在拓扑等价、blow-解析等价、Lipschitz等价、C^1等价等上寻找并引入实解析奇异的不变量，并利用不变量定理和平凡定理给出这些等价上的实解析奇异的分类。关于这些问题，我得到了以下结论：1)在与Adam Parusinski的联合工作中，我们引入了动机型不变量作为实解析函数芽的吹解析不变量，并利用我们的不变量和Fukui不变量给出了实解析函数的吹解析分类。2)我与a . parusinski一起工作过，对于2变量的实解析函数，也有不同类型的不变量。我们已经证明了牛顿边界相对于弧是一个Lipschitz不变量而牛顿边界与圆弧的点相对于弧是一个C^1不变量在两个变量的情况下。(3) Fukui不变量是众所周知的，它是真实解析函数的一种非解析不变量。另一方面，对于复全纯函数胚，存在Fukui不变量是否是拓扑不变量的问题。在2变量的情况下，我给出了肯定的答案，并构造了一个4变量函数的负例。4)对于半代数集之间的半代数映射，I和hiroshiota研究了纤维不光滑点的集合。特别是，我们寻找了沿纤维光滑部分的平凡性的不变量。然后证明了一个半代数函数沿光滑纤维是半代数平凡的，并构造了一个沿光滑纤维不平凡的半代数映射的例子。5)关于一类代数奇点的Blow-Nash模问题，我与T.Fukui和K.Bekka证明了任何多项式映射都可以表示为某些代数变体的去广化映射对该变体的严格变换与例外集在某点的交的限制。6)我引入了利普希茨弧的海缠邻域的概念，并证明了邻域的度和宽度是比利普希茨不变量，并且利用不变量证明了Briancon-Speder族和Oka族不是比利普希茨平凡。此外，在发展这一思想的基础上，我与Laurentiu Paunescu一起得到了子解析集的接吻维数是一个biLipschitz不变量的一般结果。少

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